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Theorem mapex 7860
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 6058 . . . 4 (𝑓:𝐴𝐵𝑓 ⊆ (𝐴 × 𝐵))
21ss2abi 3672 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
3 df-pw 4158 . . 3 𝒫 (𝐴 × 𝐵) = {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
42, 3sseqtr4i 3636 . 2 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5 xpexg 6957 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
6 pwexg 4848 . . 3 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
75, 6syl 17 . 2 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
8 ssexg 4802 . 2 (({𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓𝑓:𝐴𝐵} ∈ V)
94, 7, 8sylancr 695 1 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐴𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  {cab 2607  Vcvv 3198  wss 3572  𝒫 cpw 4156   × cxp 5110  wf 5882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-fun 5888  df-fn 5889  df-f 5890
This theorem is referenced by:  fnmap  7861  mapvalg  7864  isghm  17654  wksfval  26499  measbase  30245  measval  30246  ismeas  30247  isrnmeas  30248  cnfex  39013  opabresexd  41075  upwlksfval  41487
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