Theorem carduniima 4862

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 3561	. . . . 5 |- ((Fun F /\ A e. B) -> (F"A) e. V)
2		ffun 3615	. . . . 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 4830	. . . 4 |- ((F"A) e. V -> (A.x e. (F"A)(card` x) = x -> (card` U.(F"A)) = U.(F"A)))
6		ffn 3613	. . . . . . . . 9 |- (F:A-->(om u. ran aleph) -> F Fn A)
7		fnima 3590	. . . . . . . . 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 3618	. . . . . . . 8 |- (F:A-->(om u. ran aleph) -> ran F (_ (om u. ran aleph))
10	8, 9	eqsstrd 2085	. . . . . . 7 |- (F:A-->(om u. ran aleph) -> (F"A) (_ (om u. ran aleph))
11	10	sseld 2057	. . . . . 6 |- (F:A-->(om u. ran aleph) -> (x e. (F"A) -> x e. (om u. ran aleph)))
12		iscard3 4860	. . . . . 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 1705	. . . 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 4860	. 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 953   e. wcel 955  A.wral 1637  Vcvv 1802   u. cun 2035  U.cuni 2493  omcom 3121  ran crn 3161  "cima 3163  Fun wfun 3166   Fn wfn 3167  -->wf 3168  ` cfv 3172  cardccrd 4785  alephcale 4786

This theorem is referenced by:  cardinfima 4863

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

Copyright terms: Public domain