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Theorem mapex 7962
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 2735 . . 3 {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)} = {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)}
21fabexg 7959 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)} ∈ V)
3 id 22 . . . . 5 (𝑓:𝐴𝐵𝑓:𝐴𝐵)
43ancli 548 . . . 4 (𝑓:𝐴𝐵 → (𝑓:𝐴𝐵𝑓:𝐴𝐵))
54ss2abi 4077 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)}
65a1i 11 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝑓:𝐴𝐵)})
72, 6ssexd 5330 1 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  fnmap  8872  mapvalg  8875  isghmOLD  19247  wksfval  29642  measbase  34178  measval  34179  ismeas  34180  isrnmeas  34181  sticksstones4  42131  sticksstones14  42142  sticksstones20  42148  cnfex  44966  opabresexd  47237  upwlksfval  47979
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