Theorem pmex 6712

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 2312	. 2 |- { f | ( Fun f /\ f C_ ( A X. B ) ) } = { f | ( f C_ ( A X. B ) /\ Fun f ) }
3		xpexg 4777	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A X. B ) e. _V )
4		abssexg 4215	. . 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 2283	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 2167   {cab 2182   _Vcvv 2763   C_ wss 3157   X. cxp 4661   Fun wfun 5252

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-opab 4095  df-xp 4669

This theorem is referenced by: (None)