Theorem mapex 7938

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 2763	. . 3 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)}
2	1	fabexg 7936	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} ∈ V)
3		id 23	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓:𝐴⟶𝐵)
4	3	ancli 557	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵))
5	4	ss2abi 4021	. . 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 5296	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3906  ⟶wf 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-fun 6540  df-fn 6541  df-f 6542

This theorem is referenced by:  fnmap  8831  mapvalg  8834  wksfval  29937  measbase  34565  measval  34566  ismeas  34567  isrnmeas  34568  sticksstones4  42894  sticksstones14  42905  sticksstones20  42911  cnfex  45728  opabresexd  48001  upwlksfval  48877