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 ∈ C ∧ B ∈ D) → {f∣f:A–→B} ∈ 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 × B))
2	1	ss2abi 1552	. . 3 ⊢ {f∣f:A–→B} ⊆ {f∣f ⊆ (A × B)}
3		df-pw 1799	. . 3 ⊢ ℘(A × B) = {f∣f ⊆ (A × B)}
4	2, 3	sseqtr4 1533	. 2 ⊢ {f∣f:A–→B} ⊆ ℘(A × B)
5		xpexg 2489	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (A × B) ∈ V)
6		pwexg 1807	. . 3 ⊢ ((A × B) ∈ V → ℘(A × B) ∈ V)
7		ssexg 1702	. . 3 ⊢ (℘(A × B) ∈ V → ({f∣f:A–→B} ⊆ ℘(A × B) → {f∣f:A–→B} ∈ V))
8	5, 6, 7	3syl 21	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → ({f∣f:A–→B} ⊆ ℘(A × B) → {f∣f:A–→B} ∈ V))
9	4, 8	mpi 44	1 ⊢ ((A ∈ C ∧ B ∈ D) → {f∣f:A–→B} ∈ V)

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798   × 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

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