Theorem cardinfima 8920

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 3212	. 2 |- ( A e. B -> A e. _V )
2		isinfcard 8915	. . . . . . . . . . . . 13 |- ( ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) <-> ( F ` x ) e. ran aleph )
3	2	bicomi 214	. . . . . . . . . . . 12 |- ( ( F ` x ) e. ran aleph <-> ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) )
4	3	simplbi 476	. . . . . . . . . . 11 |- ( ( F ` x ) e. ran aleph -> om C_ ( F ` x ) )
5		ffn 6045	. . . . . . . . . . . 12 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnfvelrn 6356	. . . . . . . . . . . . . . . 16 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. ran F )
7	6	ex 450	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ran F ) )
8		fnima 6010	. . . . . . . . . . . . . . . 16 |- ( F Fn A -> ( F " A ) = ran F )
9	8	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( ( F ` x ) e. ( F " A ) <-> ( F ` x ) e. ran F ) )
10	7, 9	sylibrd 249	. . . . . . . . . . . . . 14 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
11		elssuni 4467	. . . . . . . . . . . . . 14 |- ( ( F ` x ) e. ( F " A ) -> ( F ` x ) C_ U. ( F " A ) )
12	10, 11	syl6 35	. . . . . . . . . . . . 13 |- ( F Fn A -> ( x e. A -> ( F ` x ) C_ U. ( F " A ) ) )
13	12	imp 445	. . . . . . . . . . . 12 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
14	5, 13	sylan 488	. . . . . . . . . . 11 |- ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
15	4, 14	sylan9ssr 3617	. . . . . . . . . 10 |- ( ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) /\ ( F ` x ) e. ran aleph ) -> om C_ U. ( F " A ) )
16	15	anasss 679	. . . . . . . . 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 8919	. . . . . . . . . 10 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )
19		iscard3 8916	. . . . . . . . . 10 |- ( ( card ` U. ( F " A ) ) = U. ( F " A ) <-> U. ( F " A ) e. ( om u. ran aleph ) )
20	18, 19	syl6ibr 242	. . . . . . . . 9 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
21	20	adantrd 484	. . . . . . . 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 555	. . . . . . 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 8915	. . . . . . 7 |- ( ( om C_ U. ( F " A ) /\ ( card ` U. ( F " A ) ) = U. ( F " A ) ) <-> U. ( F " A ) e. ran aleph )
24	22, 23	syl6ib 241	. . . . . 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 637	. . . . 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 445	. . . 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 3030	. . 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 629	. 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 17	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 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   C_ wss 3574   U.cuni 4436   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   omcom 7065   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephfplem4  8930