Theorem iunex 3869

Description: The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B(x).

Hypotheses

Ref	Expression
iunex.1	|- A e. V
iunex.2	|- B e. V

Assertion

Ref	Expression
iunex	|- U_x e. A B e. V

Distinct variable group:   x,A

Proof of Theorem iunex

Step	Hyp	Ref	Expression
1		iunex.1	. 2 |- A e. V
2		iunex.2	. . . 4 |- B e. V
3	2	a1i 8	. . 3 |- (x e. A -> B e. V)
4	3	rgen 1701	. 2 |- A.x e. A B e. V
5		iunexg 3868	. 2 |- ((A e. V /\ A.x e. A B e. V) -> U_x e. A B e. V)
6	1, 4, 5	mp2an 699	1 |- U_x e. A B e. V

Syntax hints:   e. wcel 960  A.wral 1648  Vcvv 1814  U_ciun 2570

This theorem is referenced by:  abrexex2 3877  ixpssmap 4369  tz9.1 4656  cplem2 4731

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-id 2841  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-fv 3204

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