MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapex Structured version   Visualization version   GIF version

Theorem mapex 8774
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 6697 . . . 4 (𝑓:𝐴𝐵𝑓 ⊆ (𝐴 × 𝐵))
21ss2abi 4024 . . 3 {𝑓𝑓:𝐴𝐵} ⊆ {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
3 df-pw 4563 . . 3 𝒫 (𝐴 × 𝐵) = {𝑓𝑓 ⊆ (𝐴 × 𝐵)}
42, 3sseqtrri 3982 . 2 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5 xpexg 7685 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
65pwexd 5335 . 2 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7 ssexg 5281 . 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 3444  wss 3911  𝒫 cpw 4561   × cxp 5632  wf 6493
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  fnmap  8775  mapvalg  8778  isghm  19013  permsetexOLD  19156  wksfval  28599  measbase  32853  measval  32854  ismeas  32855  isrnmeas  32856  sticksstones4  40603  sticksstones14  40614  sticksstones20  40620  cnfex  43321  opabresexd  45605  upwlksfval  46123
  Copyright terms: Public domain W3C validator