Theorem cardinfima 8368

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 3077	. 2 |- ( A e. B -> A e. _V )
2		isinfcard 8363	. . . . . . . . . . . . 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 5657	. . . . . . . . . . . 12 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnfvelrn 5939	. . . . . . . . . . . . . . . 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 5627	. . . . . . . . . . . . . . . 16 |- ( F Fn A -> ( F " A ) = ran F )
9	8	eleq2d 2521	. . . . . . . . . . . . . . 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 4219	. . . . . . . . . . . . . 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 3468	. . . . . . . . . 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 8367	. . . . . . . . . 10 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )
19		iscard3 8364	. . . . . . . . . 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 8363	. . . . . . 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 2936	. . 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 1370   e. wcel 1758   E.wrex 2796   _Vcvv 3068   u. cun 3424   C_ wss 3426   U.cuni 4189   ran crn 4939   "cima 4941   Fn wfn 5511   -->wf 5512   ` cfv 5516   omcom 6576   cardccrd 8206   alephcale 8207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-oi 7825  df-har 7874  df-card 8210  df-aleph 8211

This theorem is referenced by:  alephfplem4  8378