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Theorem mapex 7963
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 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem mapex
StepHypRef Expression
1 eqid 2737 . . 3 {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)} = {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)}
21fabexg 7960 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)} ∈ V)
3 id 22 . . . . 5 (𝑓:𝐴𝐵𝑓:𝐴𝐵)
43ancli 548 . . . 4 (𝑓:𝐴𝐵 → (𝑓:𝐴𝐵𝑓:𝐴𝐵))
54ss2abi 4067 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)}
65a1i 11 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)})
72, 6ssexd 5324 1 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  fnmap  8873  mapvalg  8876  isghmOLD  19234  wksfval  29627  measbase  34198  measval  34199  ismeas  34200  isrnmeas  34201  sticksstones4  42150  sticksstones14  42161  sticksstones20  42167  cnfex  45033  opabresexd  47299  upwlksfval  48051
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