Theorem pmex 6900

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 2350	. 2 |- { f | ( Fun f /\ f C_ ( A X. B ) ) } = { f | ( f C_ ( A X. B ) /\ Fun f ) }
3		xpexg 4869	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
4		abssexg 4300	. . 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 2321	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 2205   {cab 2220   _Vcvv 2815   C_ wss 3214   X. cxp 4752   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-opab 4177  df-xp 4760

This theorem is referenced by: (None)