Theorem pmex 5386

Description: The class of all partial functions from one set to another is a set.

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

Proof of Theorem pmex

Step	Hyp	Ref	Expression
1		xpexg 4095	. . 3 |- ((A e. C /\ B e. D) -> (A X. B) e. _V)
2		abssexg 3490	. . 3 |- ((A X. B) e. _V -> {f | (f C_ (A X. B) /\ Fun f)} e. _V)
3	1, 2	syl 12	. 2 |- ((A e. C /\ B e. D) -> {f | (f C_ (A X. B) /\ Fun f)} e. _V)
4		ancom 482	. . 3 |- ((Fun f /\ f C_ (A X. B)) <-> (f C_ (A X. B) /\ Fun f))
5	4	abbii 2006	. 2 |- {f | (Fun f /\ f C_ (A X. B))} = {f | (f C_ (A X. B) /\ Fun f)}
6	3, 5	syl5eqel 1975	1 |- ((A e. C /\ B e. D) -> {f | (Fun f /\ f C_ (A X. B))} e. _V)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593   X. cxp 3984  Fun wfun 3992

This theorem is referenced by:  pmvalg 5390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-opab 3396  df-xp 4000  df-rel 4001

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