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

Theorem funpr 6558
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
Hypotheses
Ref Expression
funpr.1 𝐴 ∈ V
funpr.2 𝐵 ∈ V
funpr.3 𝐶 ∈ V
funpr.4 𝐷 ∈ V
Assertion
Ref Expression
funpr (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})

Proof of Theorem funpr
StepHypRef Expression
1 funpr.1 . . 3 𝐴 ∈ V
2 funpr.2 . . 3 𝐵 ∈ V
31, 2pm3.2i 472 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 funpr.3 . . 3 𝐶 ∈ V
5 funpr.4 . . 3 𝐷 ∈ V
64, 5pm3.2i 472 . 2 (𝐶 ∈ V ∧ 𝐷 ∈ V)
7 funprg 6556 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
83, 6, 7mp3an12 1452 1 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2940  Vcvv 3444  {cpr 4589  cop 4593  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-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  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-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499
This theorem is referenced by:  funtp  6559  fpr  7101  fnprb  7159  1sdomOLD  9196
  Copyright terms: Public domain W3C validator