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Theorem mapex 8822
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 6742 . . . 4 (𝑓:𝐴𝐵𝑓 ⊆ (𝐴 × 𝐵))
21ss2abi 4062 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
3 df-pw 4603 . . 3 𝒫 (𝐴 × 𝐵) = {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
42, 3sseqtrri 4018 . 2 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5 xpexg 7732 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
65pwexd 5376 . 2 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7 ssexg 5322 . 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 3947  𝒫 cpw 4601   × cxp 5673  wf 6536
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544
This theorem is referenced by:  fnmap  8823  mapvalg  8826  isghm  19086  permsetexOLD  19230  wksfval  28846  measbase  33133  measval  33134  ismeas  33135  isrnmeas  33136  sticksstones4  40903  sticksstones14  40914  sticksstones20  40920  cnfex  43645  opabresexd  45930  upwlksfval  46448
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