Theorem mapex 7885

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-2003.) (Proof shortened by AV, 16-Jun-2025.)

Assertion

Ref	Expression
mapex	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem mapex

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)}
2	1	fabexg 7882	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} ∈ V)
3		id 22	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓:𝐴⟶𝐵)
4	3	ancli 548	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵))
5	4	ss2abi 4007	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)}
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)})
7	2, 6	ssexd 5261	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  {cab 2715  Vcvv 3430   ⊆ wss 3890  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fnmap  8773  mapvalg  8776  isghmOLD  19182  wksfval  29693  measbase  34357  measval  34358  ismeas  34359  isrnmeas  34360  sticksstones4  42602  sticksstones14  42613  sticksstones20  42619  cnfex  45477  opabresexd  47747  upwlksfval  48623