Theorem mapex 3261

Description: The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-03.)

Assertion

Ref	Expression
mapex	|- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Distinct variable group(s):   A,f   B,f

Proof of Theorem mapex

Step	Hyp	Ref	Expression
1		fssxp 2761	. . . 4 |- (f:A-->B -> f (_ (A X. B))
2	1	ss2abi 1552	. . 3 |- {f | f:A-->B} (_ {f | f (_ (A X. B)}
3		df-pw 1799	. . 3 |- P~(A X. B) = {f | f (_ (A X. B)}
4	2, 3	sseqtr4 1533	. 2 |- {f | f:A-->B} (_ P~(A X. B)
5		xpexg 2489	. . 3 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
6		pwexg 1807	. . 3 |- ((A X. B) e. V -> P~(A X. B) e. V)
7		ssexg 1702	. . 3 |- (P~(A X. B) e. V -> ({f | f:A-->B} (_ P~(A X. B) -> {f | f:A-->B} e. V))
8	5, 6, 7	3syl 21	. 2 |- ((A e. C /\ B e. D) -> ({f | f:A-->B} (_ P~(A X. B) -> {f | f:A-->B} e. V))
9	4, 8	mpi 44	1 |- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Syntax hints:   -> wi 2   /\ wa 196  {cab 1090   e. wcel 1092  Vcvv 1348   (_ wss 1487  P~cpw 1798   X. cxp 2408  -->wf 2418

This theorem is referenced by:  fnmap 3262  mapvalg 3263  infxpidmlem9 4941

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

metamath.org