Theorem cardiun 4842

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 3856	. . . . . 6 |- (A e. C -> {z | E.x e. A z = (card` B)} e. V)
2		visset 1810	. . . . . . . . . 10 |- y e. V
3		eqeq1 1479	. . . . . . . . . . 11 |- (z = y -> (z = (card` B) <-> y = (card` B)))
4	3	rexbidv 1662	. . . . . . . . . 10 |- (z = y -> (E.x e. A z = (card` B) <-> E.x e. A y = (card` B)))
5	2, 4	elab 1894	. . . . . . . . 9 |- (y e. {z | E.x e. A z = (card` B)} <-> E.x e. A y = (card` B))
6		cardidm 4832	. . . . . . . . . . . 12 |- (card` (card` B)) = (card` B)
7		fveq2 3719	. . . . . . . . . . . 12 |- (y = (card` B) -> (card` y) = (card` (card` B)))
8		id 59	. . . . . . . . . . . 12 |- (y = (card` B) -> y = (card` B))
9	6, 7, 8	3eqtr4a 1530	. . . . . . . . . . 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 1741	. . . . . . . . 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 1696	. . . . . . 7 |- A.y e. {z | E.x e. A z = (card` B)} (card` y) = y
14		carduni 4841	. . . . . . 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 3727	. . . . . . 7 |- (card` B) e. V
18	17	dfiun2 2583	. . . . . 6 |- U_x e. A (card` B) = U.{z | E.x e. A z = (card` B)}
19	18	fveq2i 3722	. . . . 5 |- (card` U_x e. A (card` B)) = (card` U.{z | E.x e. A z = (card` B)})
20	16, 19, 18	3eqtr4g 1529	. . . 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 2574	. . . . 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 3723	. . 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 1513	. 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 955   e. wcel 957  {cab 1462  A.wral 1643  E.wrex 1644  Vcvv 1808  U.cuni 2499  U_ciun 2562  ` cfv 3178  cardccrd 4796

This theorem is referenced by:  alephcard 4850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799

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