Theorem carduniima 6038

Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104.

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

Step	Hyp	Ref	Expression
1		funimaexg 4495	. . . . 5 |- ((Fun F /\ A e. B) -> (F"A) e. _V)
2		ffun 4565	. . . . 5 |- (F:A-->(om u. ran aleph) -> Fun F)
3	1, 2	sylan 497	. . . 4 |- ((F:A-->(om u. ran aleph) /\ A e. B) -> (F"A) e. _V)
4	3	expcom 403	. . 3 |- (A e. B -> (F:A-->(om u. ran aleph) -> (F"A) e. _V))
5		carduni 6010	. . . 4 |- ((F"A) e. _V -> (A.x e. (F"A)(card` x) = x -> (card` U.(F"A)) = U.(F"A)))
6		ffn 4562	. . . . . . . . 9 |- (F:A-->(om u. ran aleph) -> F Fn A)
7		fnima 4530	. . . . . . . . 9 |- (F Fn A -> (F"A) = ran F)
8	6, 7	syl 12	. . . . . . . 8 |- (F:A-->(om u. ran aleph) -> (F"A) = ran F)
9		frn 4569	. . . . . . . 8 |- (F:A-->(om u. ran aleph) -> ran F C_ (om u. ran aleph))
10	8, 9	eqsstrd 2651	. . . . . . 7 |- (F:A-->(om u. ran aleph) -> (F"A) C_ (om u. ran aleph))
11	10	sseld 2619	. . . . . 6 |- (F:A-->(om u. ran aleph) -> (x e. (F"A) -> x e. (om u. ran aleph)))
12		iscard3 6036	. . . . . 6 |- ((card` x) = x <-> x e. (om u. ran aleph))
13	11, 12	syl6ibr 230	. . . . 5 |- (F:A-->(om u. ran aleph) -> (x e. (F"A) -> (card` x) = x))
14	13	r19.21aiv 2175	. . . 4 |- (F:A-->(om u. ran aleph) -> A.x e. (F"A)(card` x) = x)
15	5, 14	syl5 20	. . 3 |- ((F"A) e. _V -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
16	4, 15	syli 65	. 2 |- (A e. B -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
17		iscard3 6036	. 2 |- ((card` U.(F"A)) = U.(F"A) <-> U.(F"A) e. (om u. ran aleph))
18	16, 17	syl6ib 229	1 |- (A e. B -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   u. cun 2591  U.cuni 3177  omcom 3949  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  cardccrd 5859  alephcale 5860

This theorem is referenced by:  cardinfima 6039

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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