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Theorem mapex 8398
 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 6509 . . . 4 (𝑓:𝐴𝐵𝑓 ⊆ (𝐴 × 𝐵))
21ss2abi 3994 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
3 df-pw 4499 . . 3 𝒫 (𝐴 × 𝐵) = {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
42, 3sseqtrri 3952 . 2 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5 xpexg 7456 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
65pwexd 5246 . 2 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7 ssexg 5192 . 2 (({𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓𝑓:𝐴𝐵} ∈ V)
84, 6, 7sylancr 590 1 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {cab 2776  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497   × cxp 5518  ⟶wf 6321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-fun 6327  df-fn 6328  df-f 6329 This theorem is referenced by:  fnmap  8399  mapvalg  8402  isghm  18354  permsetex  18494  wksfval  27409  measbase  31581  measval  31582  ismeas  31583  isrnmeas  31584  cnfex  41700  opabresexd  43886  upwlksfval  44406
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