Theorem cardinfima 4863

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.

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

Proof of Theorem cardinfima

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. B -> A e. V)
2		isinfcard 4859	. . . . . . . . . . . . . 14 |- ((om (_ (F` x) /\ (card` (F` x)) = (F` x)) <-> (F` x) e. ran aleph)
3	2	bicomi 172	. . . . . . . . . . . . 13 |- ((F` x) e. ran aleph <-> (om (_ (F` x) /\ (card` (F` x)) = (F` x)))
4	3	pm3.26bi 322	. . . . . . . . . . . 12 |- ((F` x) e. ran aleph -> om (_ (F` x))
5		fnfvelrn 3798	. . . . . . . . . . . . . . . . 17 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
6	5	ex 373	. . . . . . . . . . . . . . . 16 |- (F Fn A -> (x e. A -> (F` x) e. ran F))
7		fnima 3590	. . . . . . . . . . . . . . . . 17 |- (F Fn A -> (F"A) = ran F)
8	7	eleq2d 1533	. . . . . . . . . . . . . . . 16 |- (F Fn A -> ((F` x) e. (F"A) <-> (F` x) e. ran F))
9	6, 8	sylibrd 204	. . . . . . . . . . . . . . 15 |- (F Fn A -> (x e. A -> (F` x) e. (F"A)))
10		elssuni 2516	. . . . . . . . . . . . . . 15 |- ((F` x) e. (F"A) -> (F` x) (_ U.(F"A))
11	9, 10	syl6 22	. . . . . . . . . . . . . 14 |- (F Fn A -> (x e. A -> (F` x) (_ U.(F"A)))
12	11	imp 350	. . . . . . . . . . . . 13 |- ((F Fn A /\ x e. A) -> (F` x) (_ U.(F"A))
13		ffn 3613	. . . . . . . . . . . . 13 |- (F:A-->(om u. ran aleph) -> F Fn A)
14	12, 13	sylan 448	. . . . . . . . . . . 12 |- ((F:A-->(om u. ran aleph) /\ x e. A) -> (F` x) (_ U.(F"A))
15	4, 14	sylan9ssr 2066	. . . . . . . . . . 11 |- (((F:A-->(om u. ran aleph) /\ x e. A) /\ (F` x) e. ran aleph) -> om (_ U.(F"A))
16	15	anasss 440	. . . . . . . . . 10 |- ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om (_ U.(F"A))
17	16	a1i 8	. . . . . . . . 9 |- (A e. V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om (_ U.(F"A)))
18		carduniima 4862	. . . . . . . . . . 11 |- (A e. V -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))
19		iscard3 4860	. . . . . . . . . . 11 |- ((card` U.(F"A)) = U.(F"A) <-> U.(F"A) e. (om u. ran aleph))
20	18, 19	syl6ibr 213	. . . . . . . . . 10 |- (A e. V -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
21	20	adantrd 391	. . . . . . . . 9 |- (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 598	. . . . . . . 8 |- (A e. V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> (om (_ U.(F"A) /\ (card` U.(F"A)) = U.(F"A))))
23		isinfcard 4859	. . . . . . . 8 |- ((om (_ U.(F"A) /\ (card` U.(F"A)) = U.(F"A)) <-> U.(F"A) e. ran aleph)
24	22, 23	syl6ib 212	. . . . . . 7 |- (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 381	. . . . . 6 |- (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 350	. . . . 5 |- ((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	r19.23adv 1738	. . . 4 |- ((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	ex 373	. . 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)))
29	28	imp3a 361	. 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))
30	1, 29	syl 10	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 3   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638  Vcvv 1802   u. cun 2035   (_ wss 2037  U.cuni 2493  omcom 3121  ran crn 3161  "cima 3163   Fn wfn 3167  -->wf 3168  ` cfv 3172  cardccrd 4785  alephcale 4786

This theorem is referenced by:  alephfplem4 4871

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

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