Theorem infmap2 4953

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. We start with infmap2lem2 4952 and also prove the other direction of the dominance relation. We obtain equinumerosity with Schroeder-Bernstein sbth 3359 and finally eliminate the degenerate case B = ∅.

Hypotheses

Ref	Expression
infmap2.1	⊢ A ∈ V
infmap2.2	⊢ B ∈ V

Assertion

Ref	Expression
infmap2	⊢ ((ω ≼ A ∧ B ≼ A) → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})

Distinct variable group(s):   x,A   x,B

Proof of Theorem infmap2

Step	Hyp	Ref	Expression
1		infmap2.1	. . . . . . . 8 ⊢ A ∈ V
2		infmap2.2	. . . . . . . 8 ⊢ B ∈ V
3	1, 2	infxpabs 4949	. . . . . . 7 ⊢ (((ω ≼ A ∧ ¬ B = ∅) ∧ B ≼ A) → (A × B) ≈ A)
4	2, 1	xpcomen 3343	. . . . . . . 8 ⊢ (B × A) ≈ (A × B)
5		entrt 3319	. . . . . . . 8 ⊢ (((B × A) ≈ (A × B) ∧ (A × B) ≈ A) → (B × A) ≈ A)
6	4, 5	mpan 518	. . . . . . 7 ⊢ ((A × B) ≈ A → (B × A) ≈ A)
7	3, 6	syl 12	. . . . . 6 ⊢ (((ω ≼ A ∧ ¬ B = ∅) ∧ B ≼ A) → (B × A) ≈ A)
8	2, 1	xpex 2488	. . . . . . 7 ⊢ (B × A) ∈ V
9	8, 1, 2	ssenen 3399	. . . . . 6 ⊢ ((B × A) ≈ A → {x∣(x ⊆ (B × A) ∧ x ≈ B)} ≈ {x∣(x ⊆ A ∧ x ≈ B)})
10		oprex 3018	. . . . . . . 8 ⊢ (A ↑m B) ∈ V
11		abid2 1186	. . . . . . . . 9 ⊢ {x∣x ∈ (A ↑m B)} = (A ↑m B)
12	1, 2	elmap 3265	. . . . . . . . . . 11 ⊢ (x ∈ (A ↑m B) ↔ x:B–→A)
13		fssxp 2761	. . . . . . . . . . . 12 ⊢ (x:B–→A → x ⊆ (B × A))
14		ffun 2754	. . . . . . . . . . . . . 14 ⊢ (x:B–→A → Fun x)
15		visset 1350	. . . . . . . . . . . . . . 15 ⊢ x ∈ V
16	15	fundmen 3333	. . . . . . . . . . . . . 14 ⊢ (Fun x → dom x ≈ x)
17	15	ensym 3317	. . . . . . . . . . . . . 14 ⊢ (dom x ≈ x → x ≈ dom x)
18	14, 16, 17	3syl 21	. . . . . . . . . . . . 13 ⊢ (x:B–→A → x ≈ dom x)
19		fdm 2756	. . . . . . . . . . . . 13 ⊢ (x:B–→A → dom x = B)
20	18, 19	breqtrd 2081	. . . . . . . . . . . 12 ⊢ (x:B–→A → x ≈ B)
21	13, 20	jca 236	. . . . . . . . . . 11 ⊢ (x:B–→A → (x ⊆ (B × A) ∧ x ≈ B))
22	12, 21	sylbi 174	. . . . . . . . . 10 ⊢ (x ∈ (A ↑m B) → (x ⊆ (B × A) ∧ x ≈ B))
23	22	ss2abi 1552	. . . . . . . . 9 ⊢ {x∣x ∈ (A ↑m B)} ⊆ {x∣(x ⊆ (B × A) ∧ x ≈ B)}
24	11, 23	eqsstr3 1531	. . . . . . . 8 ⊢ (A ↑m B) ⊆ {x∣(x ⊆ (B × A) ∧ x ≈ B)}
25		ssdomg 3311	. . . . . . . 8 ⊢ ((A ↑m B) ∈ V → ((A ↑m B) ⊆ {x∣(x ⊆ (B × A) ∧ x ≈ B)} → (A ↑m B) ≼ {x∣(x ⊆ (B × A) ∧ x ≈ B)}))
26	10, 24, 25	mp2 43	. . . . . . 7 ⊢ (A ↑m B) ≼ {x∣(x ⊆ (B × A) ∧ x ≈ B)}
27		domentr 3326	. . . . . . 7 ⊢ (((A ↑m B) ≼ {x∣(x ⊆ (B × A) ∧ x ≈ B)} ∧ {x∣(x ⊆ (B × A) ∧ x ≈ B)} ≈ {x∣(x ⊆ A ∧ x ≈ B)}) → (A ↑m B) ≼ {x∣(x ⊆ A ∧ x ≈ B)})
28	26, 27	mpan 518	. . . . . 6 ⊢ ({x∣(x ⊆ (B × A) ∧ x ≈ B)} ≈ {x∣(x ⊆ A ∧ x ≈ B)} → (A ↑m B) ≼ {x∣(x ⊆ A ∧ x ≈ B)})
29	7, 9, 28	3syl 21	. . . . 5 ⊢ (((ω ≼ A ∧ ¬ B = ∅) ∧ B ≼ A) → (A ↑m B) ≼ {x∣(x ⊆ A ∧ x ≈ B)})
30		cleqid 1102	. . . . . . 7 ⊢ {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)} = {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)}
31	1, 2, 30	infmap2lem2 4952	. . . . . 6 ⊢ {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B)
32		sbth 3359	. . . . . 6 ⊢ (((A ↑m B) ≼ {x∣(x ⊆ A ∧ x ≈ B)} ∧ {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B)) → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})
33	31, 32	mpan2 519	. . . . 5 ⊢ ((A ↑m B) ≼ {x∣(x ⊆ A ∧ x ≈ B)} → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})
34	29, 33	syl 12	. . . 4 ⊢ (((ω ≼ A ∧ ¬ B = ∅) ∧ B ≼ A) → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})
35	34	exp31 293	. . 3 ⊢ (ω ≼ A → (¬ B = ∅ → (B ≼ A → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})))
36		opreq2 3007	. . . . . . 7 ⊢ (B = ∅ → (A ↑m B) = (A ↑m ∅))
37	1	map0e 3266	. . . . . . 7 ⊢ (A ↑m ∅) = 1o
38	36, 37	syl6eq 1140	. . . . . 6 ⊢ (B = ∅ → (A ↑m B) = 1o)
39		breq2 2066	. . . . . . . . 9 ⊢ (B = ∅ → (x ≈ B ↔ x ≈ ∅))
40	39	anbi2d 468	. . . . . . . 8 ⊢ (B = ∅ → ((x ⊆ A ∧ x ≈ B) ↔ (x ⊆ A ∧ x ≈ ∅)))
41	40	biabdv 1183	. . . . . . 7 ⊢ (B = ∅ → {x∣(x ⊆ A ∧ x ≈ B)} = {x∣(x ⊆ A ∧ x ≈ ∅)})
42		df-sn 1811	. . . . . . . 8 ⊢ {∅} = {x∣x = ∅}
43		df1o2 3111	. . . . . . . 8 ⊢ 1o = {∅}
44		en0 3328	. . . . . . . . . . 11 ⊢ (x ≈ ∅ ↔ x = ∅)
45	44	anbi2i 367	. . . . . . . . . 10 ⊢ ((x ⊆ A ∧ x ≈ ∅) ↔ (x ⊆ A ∧ x = ∅))
46		0ss 1725	. . . . . . . . . . . 12 ⊢ ∅ ⊆ A
47		sseq1 1521	. . . . . . . . . . . 12 ⊢ (x = ∅ → (x ⊆ A ↔ ∅ ⊆ A))
48	46, 47	mpbiri 169	. . . . . . . . . . 11 ⊢ (x = ∅ → x ⊆ A)
49	48	pm4.71ri 484	. . . . . . . . . 10 ⊢ (x = ∅ ↔ (x ⊆ A ∧ x = ∅))
50	45, 49	bitr4 154	. . . . . . . . 9 ⊢ ((x ⊆ A ∧ x ≈ ∅) ↔ x = ∅)
51	50	biabi 1181	. . . . . . . 8 ⊢ {x∣(x ⊆ A ∧ x ≈ ∅)} = {x∣x = ∅}
52	42, 43, 51	3eqtr4r 1127	. . . . . . 7 ⊢ {x∣(x ⊆ A ∧ x ≈ ∅)} = 1o
53	41, 52	syl6eq 1140	. . . . . 6 ⊢ (B = ∅ → {x∣(x ⊆ A ∧ x ≈ B)} = 1o)
54	38, 53	eqtr4d 1131	. . . . 5 ⊢ (B = ∅ → (A ↑m B) = {x∣(x ⊆ A ∧ x ≈ B)})
55		eqeng 3296	. . . . . 6 ⊢ ((A ↑m B) ∈ V → ((A ↑m B) = {x∣(x ⊆ A ∧ x ≈ B)} → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)}))
56	10, 55	ax-mp 6	. . . . 5 ⊢ ((A ↑m B) = {x∣(x ⊆ A ∧ x ≈ B)} → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})
57	54, 56	syl 12	. . . 4 ⊢ (B = ∅ → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})
58	57	a1d 14	. . 3 ⊢ (B = ∅ → (B ≼ A → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)}))
59	35, 58	pm2.61d2 111	. 2 ⊢ (ω ≼ A → (B ≼ A → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)}))
60	59	imp 277	1 ⊢ ((ω ≼ A ∧ B ≼ A) → (A ↑m B) ≈ {x∣(x ⊆ A ∧ x ≈ B)})

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054  {copab 2055  ωcom 2372   × cxp 2408  dom cdm 2410  Fun wfun 2416  –→wf 2418  –onto→wfo 2420  (class class class)co 3001  1_oc1o 3099   ↑_m cm 3258   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  alephexp2 4956

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-2o 3105  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-map 3259  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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