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

Theorem pmex 8773
Description: The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
Assertion
Ref Expression
pmex ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem pmex
StepHypRef Expression
1 ancom 462 . . 3 ((Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵)) ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓))
21abbii 2803 . 2 {𝑓 ∣ (Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵))} = {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)}
3 xpexg 7685 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
4 abssexg 5338 . . 3 ((𝐴 × 𝐵) ∈ V → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
53, 4syl 17 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
62, 5eqeltrid 2838 1 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (Fun 𝑓𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  {cab 2710  Vcvv 3444  wss 3911   × cxp 5632  Fun wfun 6491
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-opab 5169  df-xp 5640  df-rel 5641
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator