Theorem cardiun 4859

Description: The indexed union of a set of cardinals is a cardinal.

Assertion

Ref	Expression
cardiun	|- (A e. C -> (A.x e. A (card` B) = B -> (card` U_x e. A B) = U_x e. A B))

Distinct variable group:   x,A

Proof of Theorem cardiun

Step	Hyp	Ref	Expression
1		abrexexg 3861	. . . . . 6 |- (A e. C -> {z | E.x e. A z = (card` B)} e. V)
2		visset 1813	. . . . . . . . . 10 |- y e. V
3		eqeq1 1481	. . . . . . . . . . 11 |- (z = y -> (z = (card` B) <-> y = (card` B)))
4	3	rexbidv 1664	. . . . . . . . . 10 |- (z = y -> (E.x e. A z = (card` B) <-> E.x e. A y = (card` B)))
5	2, 4	elab 1897	. . . . . . . . 9 |- (y e. {z | E.x e. A z = (card` B)} <-> E.x e. A y = (card` B))
6		cardidm 4849	. . . . . . . . . . . 12 |- (card` (card` B)) = (card` B)
7		fveq2 3724	. . . . . . . . . . . 12 |- (y = (card` B) -> (card` y) = (card` (card` B)))
8		id 59	. . . . . . . . . . . 12 |- (y = (card` B) -> y = (card` B))
9	6, 7, 8	3eqtr4a 1532	. . . . . . . . . . 11 |- (y = (card` B) -> (card` y) = y)
10	9	a1i 8	. . . . . . . . . 10 |- (x e. A -> (y = (card` B) -> (card` y) = y))
11	10	r19.23aiv 1743	. . . . . . . . 9 |- (E.x e. A y = (card` B) -> (card` y) = y)
12	5, 11	sylbi 199	. . . . . . . 8 |- (y e. {z | E.x e. A z = (card` B)} -> (card` y) = y)
13	12	rgen 1698	. . . . . . 7 |- A.y e. {z | E.x e. A z = (card` B)} (card` y) = y
14		carduni 4858	. . . . . . 7 |- ({z | E.x e. A z = (card` B)} e. V -> (A.y e. {z | E.x e. A z = (card` B)} (card` y) = y -> (card` U.{z | E.x e. A z = (card` B)}) = U.{z | E.x e. A z = (card` B)}))
15	13, 14	mpi 44	. . . . . 6 |- ({z | E.x e. A z = (card` B)} e. V -> (card` U.{z | E.x e. A z = (card` B)}) = U.{z | E.x e. A z = (card` B)})
16	1, 15	syl 10	. . . . 5 |- (A e. C -> (card` U.{z | E.x e. A z = (card` B)}) = U.{z | E.x e. A z = (card` B)})
17		fvex 3732	. . . . . . 7 |- (card` B) e. V
18	17	dfiun2 2587	. . . . . 6 |- U_x e. A (card` B) = U.{z | E.x e. A z = (card` B)}
19	18	fveq2i 3727	. . . . 5 |- (card` U_x e. A (card` B)) = (card` U.{z | E.x e. A z = (card` B)})
20	16, 19, 18	3eqtr4g 1531	. . . 4 |- (A e. C -> (card` U_x e. A (card` B)) = U_x e. A (card` B))
21	20	adantr 389	. . 3 |- ((A e. C /\ A.x e. A (card` B) = B) -> (card` U_x e. A (card` B)) = U_x e. A (card` B))
22		iuneq2 2578	. . . . 5 |- (A.x e. A (card` B) = B -> U_x e. A (card` B) = U_x e. A B)
23	22	adantl 388	. . . 4 |- ((A e. C /\ A.x e. A (card` B) = B) -> U_x e. A (card` B) = U_x e. A B)
24	23	fveq2d 3728	. . 3 |- ((A e. C /\ A.x e. A (card` B) = B) -> (card` U_x e. A (card` B)) = (card` U_x e. A B))
25	21, 24, 23	3eqtr3d 1515	. 2 |- ((A e. C /\ A.x e. A (card` B) = B) -> (card` U_x e. A B) = U_x e. A B)
26	25	ex 373	1 |- (A e. C -> (A.x e. A (card` B) = B -> (card` U_x e. A B) = U_x e. A B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  Vcvv 1811  U.cuni 2503  U_ciun 2566  ` cfv 3182  cardccrd 4813

This theorem is referenced by:  alephcard 4867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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