Theorem mapex 7888

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 2740	. . 3 ⊢ {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)} = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵)}
2	1	fabexg 7885	. 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 553	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐵))
5	4	ss2abi 4004	. . 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 5259	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  {cab 2718  Vcvv 3432   ⊆ wss 3890  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fnmap  8777  mapvalg  8780  isghmOLD  19189  wksfval  29703  measbase  34388  measval  34389  ismeas  34390  isrnmeas  34391  sticksstones4  42641  sticksstones14  42652  sticksstones20  42658  cnfex  45483  opabresexd  47757  upwlksfval  48633