Theorem cardinfima 8467

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. (Contributed by NM, 4-Nov-2004.)

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

Allowed substitution hint:   B( x)

Proof of Theorem cardinfima

Step	Hyp	Ref	Expression
1		elex 3115	. 2 |- ( A e. B -> A e. _V )
2		isinfcard 8462	. . . . . . . . . . . . 13 |- ( ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) <-> ( F ` x ) e. ran aleph )
3	2	bicomi 202	. . . . . . . . . . . 12 |- ( ( F ` x ) e. ran aleph <-> ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) )
4	3	simplbi 460	. . . . . . . . . . 11 |- ( ( F ` x ) e. ran aleph -> om C_ ( F ` x ) )
5		ffn 5722	. . . . . . . . . . . 12 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnfvelrn 6009	. . . . . . . . . . . . . . . 16 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. ran F )
7	6	ex 434	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ran F ) )
8		fnima 5690	. . . . . . . . . . . . . . . 16 |- ( F Fn A -> ( F " A ) = ran F )
9	8	eleq2d 2530	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( ( F ` x ) e. ( F " A ) <-> ( F ` x ) e. ran F ) )
10	7, 9	sylibrd 234	. . . . . . . . . . . . . 14 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
11		elssuni 4268	. . . . . . . . . . . . . 14 |- ( ( F ` x ) e. ( F " A ) -> ( F ` x ) C_ U. ( F " A ) )
12	10, 11	syl6 33	. . . . . . . . . . . . 13 |- ( F Fn A -> ( x e. A -> ( F ` x ) C_ U. ( F " A ) ) )
13	12	imp 429	. . . . . . . . . . . 12 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
14	5, 13	sylan 471	. . . . . . . . . . 11 |- ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
15	4, 14	sylan9ssr 3511	. . . . . . . . . 10 |- ( ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) /\ ( F ` x ) e. ran aleph ) -> om C_ U. ( F " A ) )
16	15	anasss 647	. . . . . . . . 9 |- ( ( F : A --> ( om u. ran aleph ) /\ ( x e. A /\ ( F ` x ) e. ran aleph ) ) -> om C_ U. ( F " A ) )
17	16	a1i 11	. . . . . . . 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 8466	. . . . . . . . . 10 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )
19		iscard3 8463	. . . . . . . . . 10 |- ( ( card ` U. ( F " A ) ) = U. ( F " A ) <-> U. ( F " A ) e. ( om u. ran aleph ) )
20	18, 19	syl6ibr 227	. . . . . . . . 9 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
21	20	adantrd 468	. . . . . . . 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 533	. . . . . . 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 8462	. . . . . . 7 |- ( ( om C_ U. ( F " A ) /\ ( card ` U. ( F " A ) ) = U. ( F " A ) ) <-> U. ( F " A ) e. ran aleph )
24	22, 23	syl6ib 226	. . . . . 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 609	. . . . 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 429	. . . 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	rexlimdv 2946	. . 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 603	. 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 16	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 4   /\ wa 369   = wceq 1374   e. wcel 1762   E.wrex 2808   _Vcvv 3106   u. cun 3467   C_ wss 3469   U.cuni 4238   ran crn 4993   "cima 4995   Fn wfn 5574   -->wf 5575   ` cfv 5579   omcom 6671   cardccrd 8305   alephcale 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-har 7973  df-card 8309  df-aleph 8310

This theorem is referenced by:  alephfplem4  8477