Theorem cardinfima 6039

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 e. B -> ((F:A-->(om u. ran aleph) /\ E.x e. A (F` x) e. ran aleph) -> U.(F"A) e. ran aleph))

Distinct variable groups:   x,F   x,A

Proof of Theorem cardinfima

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		isinfcard 6035	. . . . . . . . . . . . 13 |- ((om C_ (F` x) /\ (card` (F` x)) = (F` x)) <-> (F` x) e. ran aleph)
3	2	bicomi 189	. . . . . . . . . . . 12 |- ((F` x) e. ran aleph <-> (om C_ (F` x) /\ (card` (F` x)) = (F` x)))
4	3	simplbi 349	. . . . . . . . . . 11 |- ((F` x) e. ran aleph -> om C_ (F` x))
5		fnfvelrn 4786	. . . . . . . . . . . . . . . 16 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
6	5	ex 402	. . . . . . . . . . . . . . 15 |- (F Fn A -> (x e. A -> (F` x) e. ran F))
7		fnima 4530	. . . . . . . . . . . . . . . 16 |- (F Fn A -> (F"A) = ran F)
8	7	eleq2d 1964	. . . . . . . . . . . . . . 15 |- (F Fn A -> ((F` x) e. (F"A) <-> (F` x) e. ran F))
9	6, 8	sylibrd 221	. . . . . . . . . . . . . 14 |- (F Fn A -> (x e. A -> (F` x) e. (F"A)))
10		elssuni 3206	. . . . . . . . . . . . . 14 |- ((F` x) e. (F"A) -> (F` x) C_ U.(F"A))
11	9, 10	syl6 25	. . . . . . . . . . . . 13 |- (F Fn A -> (x e. A -> (F` x) C_ U.(F"A)))
12	11	imp 377	. . . . . . . . . . . 12 |- ((F Fn A /\ x e. A) -> (F` x) C_ U.(F"A))
13		ffn 4562	. . . . . . . . . . . 12 |- (F:A-->(om u. ran aleph) -> F Fn A)
14	12, 13	sylan 497	. . . . . . . . . . 11 |- ((F:A-->(om u. ran aleph) /\ x e. A) -> (F` x) C_ U.(F"A))
15	4, 14	sylan9ssr 2630	. . . . . . . . . 10 |- (((F:A-->(om u. ran aleph) /\ x e. A) /\ (F` x) e. ran aleph) -> om C_ U.(F"A))
16	15	anasss 488	. . . . . . . . 9 |- ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om C_ U.(F"A))
17	16	a1i 8	. . . . . . . 8 |- (A e. _V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om C_ U.(F"A)))
18		carduniima 6038	. . . . . . . . . 10 |- (A e. _V -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))
19		iscard3 6036	. . . . . . . . . 10 |- ((card` U.(F"A)) = U.(F"A) <-> U.(F"A) e. (om u. ran aleph))
20	18, 19	syl6ibr 230	. . . . . . . . 9 |- (A e. _V -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
21	20	adantrd 427	. . . . . . . 8 |- (A e. _V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> (card` U.(F"A)) = U.(F"A)))
22	17, 21	jcad 661	. . . . . . 7 |- (A e. _V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> (om C_ U.(F"A) /\ (card` U.(F"A)) = U.(F"A))))
23		isinfcard 6035	. . . . . . 7 |- ((om C_ U.(F"A) /\ (card` U.(F"A)) = U.(F"A)) <-> U.(F"A) e. ran aleph)
24	22, 23	syl6ib 229	. . . . . 6 |- (A e. _V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> U.(F"A) e. ran aleph))
25	24	exp4d 412	. . . . 5 |- (A e. _V -> (F:A-->(om u. ran aleph) -> (x e. A -> ((F` x) e. ran aleph -> U.(F"A) e. ran aleph))))
26	25	imp 377	. . . 4 |- ((A e. _V /\ F:A-->(om u. ran aleph)) -> (x e. A -> ((F` x) e. ran aleph -> U.(F"A) e. ran aleph)))
27	26	r19.23adv 2215	. . 3 |- ((A e. _V /\ F:A-->(om u. ran aleph)) -> (E.x e. A (F` x) e. ran aleph -> U.(F"A) e. ran aleph))
28	27	expimpd 404	. 2 |- (A e. _V -> ((F:A-->(om u. ran aleph) /\ E.x e. A (F` x) e. ran aleph) -> U.(F"A) e. ran aleph))
29	1, 28	syl 12	1 |- (A e. B -> ((F:A-->(om u. ran aleph) /\ E.x e. A (F` x) e. ran aleph) -> U.(F"A) e. ran aleph))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  U.cuni 3177  omcom 3949  ran crn 3987  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  cardccrd 5859  alephcale 5860

This theorem is referenced by:  alephfplem4 6047

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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