Theorem pmex 4327

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 (_ (A X. B))} e. V)

Distinct variable groups:   A,f   B,f

Proof of Theorem pmex

Step	Hyp	Ref	Expression
1		xpexg 3259	. . 3 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
2		abssexg 2747	. . 3 |- ((A X. B) e. V -> {f | (f (_ (A X. B) /\ Fun f)} e. V)
3	1, 2	syl 10	. 2 |- ((A e. C /\ B e. D) -> {f | (f (_ (A X. B) /\ Fun f)} e. V)
4		ancom 435	. . 3 |- ((Fun f /\ f (_ (A X. B)) <-> (f (_ (A X. B) /\ Fun f))
5	4	abbii 1575	. 2 |- {f | (Fun f /\ f (_ (A X. B))} = {f | (f (_ (A X. B) /\ Fun f)}
6	3, 5	syl5eqel 1552	1 |- ((A e. C /\ B e. D) -> {f | (Fun f /\ f (_ (A X. B))} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047   X. cxp 3168  Fun wfun 3176

This theorem is referenced by:  pmvalg 4331

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-opab 2667  df-xp 3184  df-rel 3185

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