Theorem pmex 6653

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 2293	. 2 |- { f | ( Fun f /\ f C_ ( A X. B ) ) } = { f | ( f C_ ( A X. B ) /\ Fun f ) }
3		xpexg 4741	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
4		abssexg 4183	. . 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 2264	1 |- ( ( A e. C /\ B e. D ) -> { f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   {cab 2163   _Vcvv 2738   C_ wss 3130   X. cxp 4625   Fun wfun 5211

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-opab 4066  df-xp 4633

This theorem is referenced by: (None)