Theorem pmex 6739

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 2320	. 2 |- { f | ( Fun f /\ f C_ ( A X. B ) ) } = { f | ( f C_ ( A X. B ) /\ Fun f ) }
3		xpexg 4788	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
4		abssexg 4225	. . 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 2291	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 2175   {cab 2190   _Vcvv 2771   C_ wss 3165   X. cxp 4672   Fun wfun 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-opab 4105  df-xp 4680

This theorem is referenced by: (None)