Theorem cardinfima 4902

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 1820	. 2 |- (A e. B -> A e. V)
2		isinfcard 4898	. . . . . . . . . . . . 13 |- ((om (_ (F` x) /\ (card` (F` x)) = (F` x)) <-> (F` x) e. ran aleph)
3	2	bicomi 172	. . . . . . . . . . . 12 |- ((F` x) e. ran aleph <-> (om (_ (F` x) /\ (card` (F` x)) = (F` x)))
4	3	pm3.26bi 322	. . . . . . . . . . 11 |- ((F` x) e. ran aleph -> om (_ (F` x))
5		fnfvelrn 3819	. . . . . . . . . . . . . . . 16 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
6	5	ex 373	. . . . . . . . . . . . . . 15 |- (F Fn A -> (x e. A -> (F` x) e. ran F))
7		fnima 3610	. . . . . . . . . . . . . . . 16 |- (F Fn A -> (F"A) = ran F)
8	7	eleq2d 1544	. . . . . . . . . . . . . . 15 |- (F Fn A -> ((F` x) e. (F"A) <-> (F` x) e. ran F))
9	6, 8	sylibrd 204	. . . . . . . . . . . . . 14 |- (F Fn A -> (x e. A -> (F` x) e. (F"A)))
10		elssuni 2530	. . . . . . . . . . . . . 14 |- ((F` x) e. (F"A) -> (F` x) (_ U.(F"A))
11	9, 10	syl6 22	. . . . . . . . . . . . 13 |- (F Fn A -> (x e. A -> (F` x) (_ U.(F"A)))
12	11	imp 350	. . . . . . . . . . . 12 |- ((F Fn A /\ x e. A) -> (F` x) (_ U.(F"A))
13		ffn 3633	. . . . . . . . . . . 12 |- (F:A-->(om u. ran aleph) -> F Fn A)
14	12, 13	sylan 450	. . . . . . . . . . 11 |- ((F:A-->(om u. ran aleph) /\ x e. A) -> (F` x) (_ U.(F"A))
15	4, 14	sylan9ssr 2079	. . . . . . . . . 10 |- (((F:A-->(om u. ran aleph) /\ x e. A) /\ (F` x) e. ran aleph) -> om (_ U.(F"A))
16	15	anasss 442	. . . . . . . . 9 |- ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om (_ U.(F"A))
17	16	a1i 8	. . . . . . . 8 |- (A e. V -> ((F:A-->(om u. ran aleph) /\ (x e. A /\ (F` x) e. ran aleph)) -> om (_ U.(F"A)))
18		carduniima 4901	. . . . . . . . . 10 |- (A e. V -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))
19		iscard3 4899	. . . . . . . . . 10 |- ((card` U.(F"A)) = U.(F"A) <-> U.(F"A) e. (om u. ran aleph))
20	18, 19	syl6ibr 213	. . . . . . . . 9 |- (A e. V -> (F:A-->(om u. ran aleph) -> (card` U.(F"A)) = U.(F"A)))
21	20	adantrd 393	. . . . . . . 8 |- (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 602	. . . . . . 7 |- (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 4898	. . . . . . 7 |- ((om (_ U.(F"A) /\ (card` U.(F"A)) = U.(F"A)) <-> U.(F"A) e. ran aleph)
24	22, 23	syl6ib 212	. . . . . 6 |- (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 383	. . . . 5 |- (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	. . . 4 |- ((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 1749	. . 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))
28	27	expimpd 375	. 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))
29	1, 28	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 958   e. wcel 960  E.wrex 1649  Vcvv 1814   u. cun 2048   (_ wss 2050  U.cuni 2507  omcom 3137  ran crn 3177  "cima 3179   Fn wfn 3183  -->wf 3184  ` cfv 3188  cardccrd 4823  alephcale 4824

This theorem is referenced by:  alephfplem4 4910

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827

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