Theorem cardinfima 8546

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 3040	. 2 |- ( A e. B -> A e. _V )
2		isinfcard 8541	. . . . . . . . . . . . 13 |- ( ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) <-> ( F ` x ) e. ran aleph )
3	2	bicomi 207	. . . . . . . . . . . 12 |- ( ( F ` x ) e. ran aleph <-> ( om C_ ( F ` x ) /\ ( card ` ( F ` x ) ) = ( F ` x ) ) )
4	3	simplbi 467	. . . . . . . . . . 11 |- ( ( F ` x ) e. ran aleph -> om C_ ( F ` x ) )
5		ffn 5739	. . . . . . . . . . . 12 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnfvelrn 6034	. . . . . . . . . . . . . . . 16 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. ran F )
7	6	ex 441	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ran F ) )
8		fnima 5704	. . . . . . . . . . . . . . . 16 |- ( F Fn A -> ( F " A ) = ran F )
9	8	eleq2d 2534	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( ( F ` x ) e. ( F " A ) <-> ( F ` x ) e. ran F ) )
10	7, 9	sylibrd 242	. . . . . . . . . . . . . 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 436	. . . . . . . . . . . 12 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
14	5, 13	sylan 479	. . . . . . . . . . 11 |- ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
15	4, 14	sylan9ssr 3432	. . . . . . . . . 10 |- ( ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) /\ ( F ` x ) e. ran aleph ) -> om C_ U. ( F " A ) )
16	15	anasss 659	. . . . . . . . 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 8545	. . . . . . . . . 10 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )
19		iscard3 8542	. . . . . . . . . 10 |- ( ( card ` U. ( F " A ) ) = U. ( F " A ) <-> U. ( F " A ) e. ( om u. ran aleph ) )
20	18, 19	syl6ibr 235	. . . . . . . . 9 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
21	20	adantrd 475	. . . . . . . 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 542	. . . . . . 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 8541	. . . . . . 7 |- ( ( om C_ U. ( F " A ) /\ ( card ` U. ( F " A ) ) = U. ( F " A ) ) <-> U. ( F " A ) e. ran aleph )
24	22, 23	syl6ib 234	. . . . . 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 620	. . . . 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 436	. . . 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 2870	. . 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 614	. 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 376   = wceq 1452   e. wcel 1904   E.wrex 2757   _Vcvv 3031   u. cun 3388   C_ wss 3390   U.cuni 4190   ran crn 4840   "cima 4842   Fn wfn 5584   -->wf 5585   ` cfv 5589   omcom 6711   cardccrd 8387   alephcale 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392

This theorem is referenced by:  alephfplem4  8556