Theorem cardinfima 8469

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 8464	. . . . . . . . . . . . 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 458	. . . . . . . . . . 11 |- ( ( F ` x ) e. ran aleph -> om C_ ( F ` x ) )
5		ffn 5713	. . . . . . . . . . . 12 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnfvelrn 6004	. . . . . . . . . . . . . . . 16 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. ran F )
7	6	ex 432	. . . . . . . . . . . . . . 15 |- ( F Fn A -> ( x e. A -> ( F ` x ) e. ran F ) )
8		fnima 5681	. . . . . . . . . . . . . . . 16 |- ( F Fn A -> ( F " A ) = ran F )
9	8	eleq2d 2524	. . . . . . . . . . . . . . 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 4264	. . . . . . . . . . . . . 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 427	. . . . . . . . . . . 12 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
14	5, 13	sylan 469	. . . . . . . . . . 11 |- ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) -> ( F ` x ) C_ U. ( F " A ) )
15	4, 14	sylan9ssr 3503	. . . . . . . . . 10 |- ( ( ( F : A --> ( om u. ran aleph ) /\ x e. A ) /\ ( F ` x ) e. ran aleph ) -> om C_ U. ( F " A ) )
16	15	anasss 645	. . . . . . . . 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 8468	. . . . . . . . . 10 |- ( A e. _V -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )
19		iscard3 8465	. . . . . . . . . 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 466	. . . . . . . 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 531	. . . . . . 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 8464	. . . . . . 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 607	. . . . 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 427	. . . 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 2944	. . 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 601	. 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 367   = wceq 1398   e. wcel 1823   E.wrex 2805   _Vcvv 3106   u. cun 3459   C_ wss 3461   U.cuni 4235   ran crn 4989   "cima 4991   Fn wfn 5565   -->wf 5566   ` cfv 5570   omcom 6673   cardccrd 8307   alephcale 8308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312

This theorem is referenced by:  alephfplem4  8479