Theorem iunex 6342

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 ). (Contributed by NM, 13-Oct-2003.)

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

Allowed substitution hint:   B( x)

Proof of Theorem iunex

Step	Hyp	Ref	Expression
1		iunex.1	. 2 |- A e. _V
2		iunex.2	. . 3 |- B e. _V
3	2	rgenw 2605	. 2 |- A. x e. A B e. _V
4		iunexg 6338	. 2 |- ( ( A e. _V /\ A. x e. A B e. _V ) -> U_ x e. A B e. _V )
5	1, 3, 4	mp2an 430	1 |- U_ x e. A B e. _V

Syntax hints:   e. wcel 2209   A.wral 2528   _Vcvv 2821   U_ciun 4007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by:  abrexex2  6343