Theorem pmex 6927

Description: The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Assertion

Ref	Expression
pmex	|- ( ( A e. C /\ B e. D ) -> { f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V )

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   C( f)   D( f)

Proof of Theorem pmex

Step	Hyp	Ref	Expression
1		ancom 266	. . 3 |- ( ( Fun f /\ f C_ ( A X. B ) ) <-> ( f C_ ( A X. B ) /\ Fun f ) )
2	1	abbii 2354	. 2 |- { f | ( Fun f /\ f C_ ( A X. B ) ) } = { f | ( f C_ ( A X. B ) /\ Fun f ) }
3		xpexg 4889	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
4		abssexg 4319	. . 3 |- ( ( A X. B ) e. _V -> { f | ( f C_ ( A X. B ) /\ Fun f ) } e. _V )
5	3, 4	syl 14	. 2 |- ( ( A e. C /\ B e. D ) -> { f | ( f C_ ( A X. B ) /\ Fun f ) } e. _V )
6	2, 5	eqeltrid 2325	1 |- ( ( A e. C /\ B e. D ) -> { f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   {cab 2224   _Vcvv 2821   C_ wss 3220   X. cxp 4772   Fun wfun 5371

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-opab 4193  df-xp 4780

This theorem is used by: (None)