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Theorem mapex 8826
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.)
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 fssxp 6746 . . . 4 (𝑓:𝐴𝐵𝑓 ⊆ (𝐴 × 𝐵))
21ss2abi 4064 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
3 df-pw 4605 . . 3 𝒫 (𝐴 × 𝐵) = {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
42, 3sseqtrri 4020 . 2 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5 xpexg 7737 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
65pwexd 5378 . 2 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7 ssexg 5324 . 2 (({𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓𝑓:𝐴𝐵} ∈ V)
84, 6, 7sylancr 588 1 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  {cab 2710  Vcvv 3475  wss 3949  𝒫 cpw 4603   × cxp 5675  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  fnmap  8827  mapvalg  8830  isghm  19092  permsetexOLD  19237  wksfval  28866  measbase  33195  measval  33196  ismeas  33197  isrnmeas  33198  sticksstones4  40965  sticksstones14  40976  sticksstones20  40982  cnfex  43712  opabresexd  45995  upwlksfval  46513
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