Theorem infmap2lem2 4952

Description: Lemma for infmap2 4953. Given the relation R, we use the Axiom of Choice ac7g 3570 to extract a function f that provides the one-to-one mapping for the dominance relation.

Hypotheses

Ref	Expression
infmap2lem.1	⊢ A ∈ V
infmap2lem.2	⊢ B ∈ V
infmap2lem.3	⊢ R = {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)}

Assertion

Ref	Expression
infmap2lem2	⊢ {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B)

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

Proof of Theorem infmap2lem2

Step	Hyp	Ref	Expression
1		infmap2lem.3	. . . 4 ⊢ R = {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)}
2		df-xp 2424	. . . . . 6 ⊢ (℘A × (A ↑m B)) = {⟨z, w⟩∣(z ∈ ℘A ∧ w ∈ (A ↑m B))}
3		infmap2lem.1	. . . . . . . 8 ⊢ A ∈ V
4	3	pwex 1806	. . . . . . 7 ⊢ ℘A ∈ V
5		oprex 3018	. . . . . . 7 ⊢ (A ↑m B) ∈ V
6	4, 5	xpex 2488	. . . . . 6 ⊢ (℘A × (A ↑m B)) ∈ V
7	2, 6	eqeltrr 1160	. . . . 5 ⊢ {⟨z, w⟩∣(z ∈ ℘A ∧ w ∈ (A ↑m B))} ∈ V
8		pm3.26 256	. . . . . . . . 9 ⊢ ((z ⊆ A ∧ w:B–onto→z) → z ⊆ A)
9		visset 1350	. . . . . . . . . 10 ⊢ z ∈ V
10	9	elpw 1801	. . . . . . . . 9 ⊢ (z ∈ ℘A ↔ z ⊆ A)
11	8, 10	sylibr 175	. . . . . . . 8 ⊢ ((z ⊆ A ∧ w:B–onto→z) → z ∈ ℘A)
12		fof 2788	. . . . . . . . . . . 12 ⊢ (w:B–onto→z → w:B–→z)
13		ffn 2752	. . . . . . . . . . . 12 ⊢ (w:B–→z → w Fn B)
14	12, 13	syl 12	. . . . . . . . . . 11 ⊢ (w:B–onto→z → w Fn B)
15	14	adantl 305	. . . . . . . . . 10 ⊢ ((z ⊆ A ∧ w:B–onto→z) → w Fn B)
16		forn 2789	. . . . . . . . . . . 12 ⊢ (w:B–onto→z → ran w = z)
17	16	sseq1d 1527	. . . . . . . . . . 11 ⊢ (w:B–onto→z → (ran w ⊆ A ↔ z ⊆ A))
18	17	biimparc 327	. . . . . . . . . 10 ⊢ ((z ⊆ A ∧ w:B–onto→z) → ran w ⊆ A)
19	15, 18	jca 236	. . . . . . . . 9 ⊢ ((z ⊆ A ∧ w:B–onto→z) → (w Fn B ∧ ran w ⊆ A))
20		infmap2lem.2	. . . . . . . . . . 11 ⊢ B ∈ V
21	3, 20	elmap 3265	. . . . . . . . . 10 ⊢ (w ∈ (A ↑m B) ↔ w:B–→A)
22		df-f 2434	. . . . . . . . . 10 ⊢ (w:B–→A ↔ (w Fn B ∧ ran w ⊆ A))
23	21, 22	bitr 151	. . . . . . . . 9 ⊢ (w ∈ (A ↑m B) ↔ (w Fn B ∧ ran w ⊆ A))
24	19, 23	sylibr 175	. . . . . . . 8 ⊢ ((z ⊆ A ∧ w:B–onto→z) → w ∈ (A ↑m B))
25	11, 24	jca 236	. . . . . . 7 ⊢ ((z ⊆ A ∧ w:B–onto→z) → (z ∈ ℘A ∧ w ∈ (A ↑m B)))
26	25	adantlr 310	. . . . . 6 ⊢ (((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z) → (z ∈ ℘A ∧ w ∈ (A ↑m B)))
27	26	ssopab2i 2120	. . . . 5 ⊢ {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)} ⊆ {⟨z, w⟩∣(z ∈ ℘A ∧ w ∈ (A ↑m B))}
28	7, 27	ssexi 1701	. . . 4 ⊢ {⟨z, w⟩∣((z ⊆ A ∧ z ≈ B) ∧ w:B–onto→z)} ∈ V
29	1, 28	eqeltr 1159	. . 3 ⊢ R ∈ V
30		ac7g 3570	. . 3 ⊢ (R ∈ V → ∃f(f ⊆ R ∧ f Fn dom R))
31	29, 30	ax-mp 6	. 2 ⊢ ∃f(f ⊆ R ∧ f Fn dom R)
32		df-pw 1799	. . . . . 6 ⊢ ℘A = {x∣x ⊆ A}
33	32, 4	eqeltrr 1160	. . . . 5 ⊢ {x∣x ⊆ A} ∈ V
34		pm3.26 256	. . . . . 6 ⊢ ((x ⊆ A ∧ x ≈ B) → x ⊆ A)
35	34	ss2abi 1552	. . . . 5 ⊢ {x∣(x ⊆ A ∧ x ≈ B)} ⊆ {x∣x ⊆ A}
36	33, 35	ssexi 1701	. . . 4 ⊢ {x∣(x ⊆ A ∧ x ≈ B)} ∈ V
37	3, 20, 1	infmap2lem1 4951	. . . . . 6 ⊢ ((f ⊆ R ∧ f Fn dom R) → (v ∈ {x∣(x ⊆ A ∧ x ≈ B)} → (v ⊆ A ∧ (f ‘v):B–onto→v)))
38		fss 2759	. . . . . . . . 9 ⊢ (((f ‘v):B–→v ∧ v ⊆ A) → (f ‘v):B–→A)
39		fof 2788	. . . . . . . . 9 ⊢ ((f ‘v):B–onto→v → (f ‘v):B–→v)
40	38, 39	sylan 343	. . . . . . . 8 ⊢ (((f ‘v):B–onto→v ∧ v ⊆ A) → (f ‘v):B–→A)
41	40	ancoms 334	. . . . . . 7 ⊢ ((v ⊆ A ∧ (f ‘v):B–onto→v) → (f ‘v):B–→A)
42	3, 20	elmap 3265	. . . . . . 7 ⊢ ((f ‘v) ∈ (A ↑m B) ↔ (f ‘v):B–→A)
43	41, 42	sylibr 175	. . . . . 6 ⊢ ((v ⊆ A ∧ (f ‘v):B–onto→v) → (f ‘v) ∈ (A ↑m B))
44	37, 43	syl6 23	. . . . 5 ⊢ ((f ⊆ R ∧ f Fn dom R) → (v ∈ {x∣(x ⊆ A ∧ x ≈ B)} → (f ‘v) ∈ (A ↑m B)))
45		pm3.27 260	. . . . . . . 8 ⊢ ((v ⊆ A ∧ (f ‘v):B–onto→v) → (f ‘v):B–onto→v)
46	37, 45	syl6 23	. . . . . . 7 ⊢ ((f ⊆ R ∧ f Fn dom R) → (v ∈ {x∣(x ⊆ A ∧ x ≈ B)} → (f ‘v):B–onto→v))
47	3, 20, 1	infmap2lem1 4951	. . . . . . . 8 ⊢ ((f ⊆ R ∧ f Fn dom R) → (u ∈ {x∣(x ⊆ A ∧ x ≈ B)} → (u ⊆ A ∧ (f ‘u):B–onto→u)))
48		pm3.27 260	. . . . . . . 8 ⊢ ((u ⊆ A ∧ (f ‘u):B–onto→u) → (f ‘u):B–onto→u)
49	47, 48	syl6 23	. . . . . . 7 ⊢ ((f ⊆ R ∧ f Fn dom R) → (u ∈ {x∣(x ⊆ A ∧ x ≈ B)} → (f ‘u):B–onto→u))
50	46, 49	anim12d 431	. . . . . 6 ⊢ ((f ⊆ R ∧ f Fn dom R) → ((v ∈ {x∣(x ⊆ A ∧ x ≈ B)} ∧ u ∈ {x∣(x ⊆ A ∧ x ≈ B)}) → ((f ‘v):B–onto→v ∧ (f ‘u):B–onto→u)))
51		forn 2789	. . . . . . . . 9 ⊢ ((f ‘v):B–onto→v → ran (f ‘v) = v)
52		forn 2789	. . . . . . . . 9 ⊢ ((f ‘u):B–onto→u → ran (f ‘u) = u)
53	51, 52	cleqan12d 1116	. . . . . . . 8 ⊢ (((f ‘v):B–onto→v ∧ (f ‘u):B–onto→u) → (ran (f ‘v) = ran (f ‘u) ↔ v = u))
54		rneq 2555	. . . . . . . 8 ⊢ ((f ‘v) = (f ‘u) → ran (f ‘v) = ran (f ‘u))
55	53, 54	syl5bi 183	. . . . . . 7 ⊢ (((f ‘v):B–onto→v ∧ (f ‘u):B–onto→u) → ((f ‘v) = (f ‘u) → v = u))
56		fveq2 2832	. . . . . . . 8 ⊢ (v = u → (f ‘v) = (f ‘u))
57	56	a1i 7	. . . . . . 7 ⊢ (((f ‘v):B–onto→v ∧ (f ‘u):B–onto→u) → (v = u → (f ‘v) = (f ‘u)))
58	55, 57	impbid 397	. . . . . 6 ⊢ (((f ‘v):B–onto→<Ä>v ∧ (f ‘u):B–onto→u) → ((f ‘v) = (f ‘u) ↔ v = u))
59	50, 58	syl6 23	. . . . 5 ⊢ ((f ⊆ R ∧ f Fn dom R) → ((v ∈ {x∣(x ⊆ A ∧ x ≈ B)} ∧ u ∈ {x∣(x ⊆ A ∧ x ≈ B)}) → ((f ‘v) = (f ‘u) ↔ v = u)))
60	44, 59	dom2d 3307	. . . 4 ⊢ ((f ⊆ R ∧ f Fn dom R) → ({x∣(x ⊆ A ∧ x ≈ B)} ∈ V → {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B)))
61	36, 60	mpi 44	. . 3 ⊢ ((f ⊆ R ∧ f Fn dom R) → {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B))
62	61	19.23aiv 952	. 2 ⊢ (∃f(f ⊆ R ∧ f Fn dom R) → {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B))
63	31, 62	ax-mp 6	1 ⊢ {x∣(x ⊆ A ∧ x ≈ B)} ≼ (A ↑m B)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798   class class class wbr 2054  {copab 2055   × cxp 2408  dom cdm 2410  ran crn 2411   Fn wfn 2417  –→wf 2418  –onto→wfo 2420   ‘cfv 2422  (class class class)co 3001   ↑_m cm 3258   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  infmap2 4953

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-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  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-opr 3003  df-oprab 3004  df-er 3200  df-map 3259  df-en 3274  df-dom 3275

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