Theorem cardinfima 3696

Description: If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104.

Assertion

Ref	Expression
cardinfima	⊢ (A ∈ B → ((F:A–→(ω ∪ ran ℵ) ∧ ∃x ∈ A (F ‘x) ∈ ran ℵ) → ∪(F “ A) ∈ ran ℵ))

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

Proof of Theorem cardinfima

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (A ∈ B → A ∈ V)
2		isinfcard 3692	. . . . . . . . . . . . . 14 ⊢ ((ω ⊆ (F ‘x) ∧ (card ‘(F ‘x)) = (F ‘x)) ↔ (F ‘x) ∈ ran ℵ)
3	2	bicomi 150	. . . . . . . . . . . . 13 ⊢ ((F ‘x) ∈ ran ℵ ↔ (ω ⊆ (F ‘x) ∧ (card ‘(F ‘x)) = (F ‘x)))
4	3	pm3.26bd 259	. . . . . . . . . . . 12 ⊢ ((F ‘x) ∈ ran ℵ → ω ⊆ (F ‘x))
5		fnfvrn 2889	. . . . . . . . . . . . . . . . 17 ⊢ ((F Fn A ∧ x ∈ A) → (F ‘x) ∈ ran F)
6	5	exp 291	. . . . . . . . . . . . . . . 16 ⊢ (F Fn A → (x ∈ A → (F ‘x) ∈ ran F))
7		fnima 2738	. . . . . . . . . . . . . . . . 17 ⊢ (F Fn A → (F “ A) = ran F)
8	7	eleq2d 1156	. . . . . . . . . . . . . . . 16 ⊢ (F Fn A → ((F ‘x) ∈ (F “ A) ↔ (F ‘x) ∈ ran F))
9	6, 8	sylibrd 179	. . . . . . . . . . . . . . 15 ⊢ (F Fn A → (x ∈ A → (F ‘x) ∈ (F “ A)))
10		elssuni 1940	. . . . . . . . . . . . . . 15 ⊢ ((F ‘x) ∈ (F “ A) → (F ‘x) ⊆ ∪(F “ A))
11	9, 10	syl6 23	. . . . . . . . . . . . . 14 ⊢ (F Fn A → (x ∈ A → (F ‘x) ⊆ ∪(F “ A)))
12	11	imp 277	. . . . . . . . . . . . 13 ⊢ ((F Fn A ∧ x ∈ A) → (F ‘x) ⊆ ∪(F “ A))
13		ffn 2752	. . . . . . . . . . . . 13 ⊢ (F:A–→(ω ∪ ran ℵ) → F Fn A)
14	12, 13	sylan 343	. . . . . . . . . . . 12 ⊢ ((F:A–→(ω ∪ ran ℵ) ∧ x ∈ A) → (F ‘x) ⊆ ∪(F “ A))
15	4, 14	sylan9ssr 1515	. . . . . . . . . . 11 ⊢ (((F:A–→(ω ∪ ran ℵ) ∧ x ∈ A) ∧ (F ‘x) ∈ ran ℵ) → ω ⊆ ∪(F “ A))
16	15	anasss 337	. . . . . . . . . 10 ⊢ ((F:A–→(ω ∪ ran ℵ) ∧ (x ∈ A ∧ (F ‘x) ∈ ran ℵ)) → ω ⊆ ∪(F “ A))
17	16	a1i 7	. . . . . . . . 9 ⊢ (A ∈ V → ((F:A–→(ω ∪ ran ℵ) ∧ (x ∈ A ∧ (F ‘x) ∈ ran ℵ)) → ω ⊆ ∪(F “ A)))
18		carduniima 3695	. . . . . . . . . . 11 ⊢ (A ∈ V → (F:A–→(ω ∪ ran ℵ) → ∪(F “ A) ∈ (ω ∪ ran ℵ)))
19		iscard3 3693	. . . . . . . . . . 11 ⊢ ((card ‘∪(F “ A)) = ∪(F “ A) ↔ ∪(F “ A) ∈ (ω ∪ ran ℵ))
20	18, 19	syl6ibr 186	. . . . . . . . . 10 ⊢ (A ∈ V → (F:A–→(ω ∪ ran ℵ) → (card ‘∪(F “ A)) = ∪(F “ A)))
21	20	adantrd 308	. . . . . . . . 9 ⊢ (A ∈ V → ((F:A–→(ω ∪ ran ℵ) ∧ (x ∈ A ∧ (F ‘x) ∈ ran ℵ)) → (card ‘∪(F “ A)) = ∪(F “ A)))
22	17, 21	jcad 455	. . . . . . . 8 ⊢ (A ∈ V → ((F:A–→(ω ∪ ran ℵ) ∧ (x ∈ A ∧ (F ‘x) ∈ ran ℵ)) → (ω ⊆ ∪(F “ A) ∧ (card ‘∪(F “ A)) = ∪(F “ A))))
23		isinfcard 3692	. . . . . . . 8 ⊢ ((ω ⊆ ∪(F “ A) ∧ (card ‘∪(F “ A)) = ∪(F “ A)) ↔ ∪(F “ A) ∈ ran ℵ)
24	22, 23	syl6ib 185	. . . . . . 7 ⊢ (A ∈ V → ((F:A–→(ω ∪ ran ℵ) ∧ (x ∈ A ∧ (F ‘x) ∈ ran ℵ)) → ∪(F “ A) ∈ ran ℵ))
25	24	exp4d 298	. . . . . 6 ⊢ (A ∈ V → (F:A–→(ω ∪ ran ℵ) → (x ∈ A → ((F ‘x) ∈ ran ℵ → ∪(F “ A) ∈ ran ℵ))))
26	25	imp 277	. . . . 5 ⊢ ((A ∈ V ∧ F:A–→(ω ∪ ran ℵ)) → (x ∈ A → ((F ‘x) ∈ ran ℵ → ∪(F “ A) ∈ ran ℵ)))
27	26	r19.23adv 1286	. . . 4 ⊢ ((A ∈ V ∧ F:A–→(ω ∪ ran ℵ)) → (∃x ∈ A (F ‘x) ∈ ran ℵ → ∪(F “ A) ∈ ran ℵ))
28	27	exp 291	. . 3 ⊢ (A ∈ V → (F:A–→(ω ∪ ran ℵ) → (∃x ∈ A (F ‘x) ∈ ran ℵ → ∪(F “ A) ∈ ran ℵ)))
29	28	imp3a 279	. 2 ⊢ (A ∈ V → ((F:A–→(ω ∪ ran ℵ) ∧ ∃x ∈ A (F ‘x) ∈ ran ℵ) → ∪(F “ A) ∈ ran ℵ))
30	1, 29	syl 12	1 ⊢ (A ∈ B → ((F:A–→(ω ∪ ran ℵ) ∧ ∃x ∈ A (F ‘x) ∈ ran ℵ) → ∪(F “ A) ∈ ran ℵ))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487  ∪cuni 1919  ωcom 2372  ran crn 2411   “ cima 2413   Fn wfn 2417  –→wf 2418   ‘cfv 2422  cardccrd 3620  ℵcale 3621

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-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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