Theorem carduniima 4870

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 3567	. . . . 5 |- ((Fun F /\ A e. B) -> (F"A) e. V)
2		ffun 3621	. . . . 5 |- (F:A-->(om u. ran aleph) -> Fun F)
3	1, 2	sylan 448	. . . 4 |- ((F:A-->(om u. ran aleph) /\ A e. B) -> (F"A) e. V)
4	3	expcom 374	. . 3 |- (A e. B -> (F:A-->(om u. ran aleph) -> (F"A) e. V))
5		carduni 4838	. . . 4 |- ((F"A) e. V -> (A.x e. (F"A)(card` x) = x -> (card` U.(F"A)) = U.(F"A)))
6		ffn 3619	. . . . . . . . 9 |- (F:A-->(om u. ran aleph) -> F Fn A)
7		fnima 3596	. . . . . . . . 9 |- (F Fn A -> (F"A) = ran F)
8	6, 7	syl 10	. . . . . . . 8 |- (F:A-->(om u. ran aleph) -> (F"A) = ran F)
9		frn 3624	. . . . . . . 8 |- (F:A-->(om u. ran aleph) -> ran F (_ (om u. ran aleph))
10	8, 9	eqsstrd 2091	. . . . . . 7 |- (F:A-->(om u. ran aleph) -> (F"A) (_ (om u. ran aleph))
11	10	sseld 2063	. . . . . 6 |- (F:A-->(om u. ran aleph) -> (x e. (F"A) -> x e. (om u. ran aleph)))
12		iscard3 4868	. . . . . 6 |- ((card` x) = x <-> x e. (om u. ran aleph))
13	11, 12	syl6ibr 213	. . . . 5 |- (F:A-->(om u. ran aleph) -> (x e. (F"A) -> (card` x) = x))
14	13	r19.21aiv 1710	. . . 4 |- (F:A-->(om u. ran aleph) -> A.x e. (F"A)(card` x) = x)
15	5, 14	syl5 21	. . 3 |- ((F"A) e. V -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
16	4, 15	syli 54	. 2 |- (A e. B -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
17		iscard3 4868	. 2 |- ((card` U.(F"A)) = U.(F"A) <-> U.(F"A) e. (om u. ran aleph))
18	16, 17	syl6ib 212	1 |- (A e. B -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807   u. cun 2041  U.cuni 2498  omcom 3126  ran crn 3166  "cima 3168  Fun wfun 3171   Fn wfn 3172  -->wf 3173  ` cfv 3177  cardccrd 4793  alephcale 4794

This theorem is referenced by:  cardinfima 4871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796  df-aleph 4797

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