Theorem fabex 5578

Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Hypotheses

Ref	Expression
fabex.1	|- A e. _V
fabex.2	|- B e. _V
fabex.3	|- F = { x | ( x : A --> B /\ ph ) }

Assertion

Ref	Expression
fabex	|- F e. _V

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   F( x)

Proof of Theorem fabex

Step	Hyp	Ref	Expression
1		fabex.1	. 2 |- A e. _V
2		fabex.2	. 2 |- B e. _V
3		fabex.3	. . 3 |- F = { x | ( x : A --> B /\ ph ) }
4	3	fabexg 5577	. 2 |- ( ( A e. _V /\ B e. _V ) -> F e. _V )
5	1, 2, 4	mp2an 430	1 |- F e. _V

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   -->wf 5371

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-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

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-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-dm 4782  df-rn 4783  df-fun 5377  df-fn 5378  df-f 5379

This theorem is referenced by: (None)