Theorem carduniima 8919

Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)

Assertion

Ref	Expression
carduniima	|- ( A e. B -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )

Proof of Theorem carduniima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffun 6048	. . . . 5 |- ( F : A --> ( om u. ran aleph ) -> Fun F )
2		funimaexg 5975	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
3	1, 2	sylan 488	. . . 4 |- ( ( F : A --> ( om u. ran aleph ) /\ A e. B ) -> ( F " A ) e. _V )
4	3	expcom 451	. . 3 |- ( A e. B -> ( F : A --> ( om u. ran aleph ) -> ( F " A ) e. _V ) )
5		ffn 6045	. . . . . . . . 9 |- ( F : A --> ( om u. ran aleph ) -> F Fn A )
6		fnima 6010	. . . . . . . . 9 |- ( F Fn A -> ( F " A ) = ran F )
7	5, 6	syl 17	. . . . . . . 8 |- ( F : A --> ( om u. ran aleph ) -> ( F " A ) = ran F )
8		frn 6053	. . . . . . . 8 |- ( F : A --> ( om u. ran aleph ) -> ran F C_ ( om u. ran aleph ) )
9	7, 8	eqsstrd 3639	. . . . . . 7 |- ( F : A --> ( om u. ran aleph ) -> ( F " A ) C_ ( om u. ran aleph ) )
10	9	sseld 3602	. . . . . 6 |- ( F : A --> ( om u. ran aleph ) -> ( x e. ( F " A ) -> x e. ( om u. ran aleph ) ) )
11		iscard3 8916	. . . . . 6 |- ( ( card ` x ) = x <-> x e. ( om u. ran aleph ) )
12	10, 11	syl6ibr 242	. . . . 5 |- ( F : A --> ( om u. ran aleph ) -> ( x e. ( F " A ) -> ( card ` x ) = x ) )
13	12	ralrimiv 2965	. . . 4 |- ( F : A --> ( om u. ran aleph ) -> A. x e. ( F " A ) ( card ` x ) = x )
14		carduni 8807	. . . 4 |- ( ( F " A ) e. _V -> ( A. x e. ( F " A ) ( card ` x ) = x -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
15	13, 14	syl5 34	. . 3 |- ( ( F " A ) e. _V -> ( F : A --> ( om u. ran aleph ) -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
16	4, 15	syli 39	. 2 |- ( A e. B -> ( F : A --> ( om u. ran aleph ) -> ( card ` U. ( F " A ) ) = U. ( F " A ) ) )
17		iscard3 8916	. 2 |- ( ( card ` U. ( F " A ) ) = U. ( F " A ) <-> U. ( F " A ) e. ( om u. ran aleph ) )
18	16, 17	syl6ib 241	1 |- ( A e. B -> ( F : A --> ( om u. ran aleph ) -> U. ( F " A ) e. ( om u. ran aleph ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   U.cuni 4436   ran crn 5115   "cima 5117   Fun wfun 5882   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:  cardinfima  8920