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

Theorem funpr 6182
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 464 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 funpr.3 . . 3 𝐶 ∈ V
5 funpr.4 . . 3 𝐷 ∈ V
64, 5pm3.2i 464 . 2 (𝐶 ∈ V ∧ 𝐷 ∈ V)
7 funprg 6180 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
83, 6, 7mp3an12 1579 1 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  wne 2999  Vcvv 3414  {cpr 4401  cop 4405  Fun wfun 6121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-fun 6129
This theorem is referenced by:  funtp  6183  fpr  6677  fnprb  6733  1sdom  8438
  Copyright terms: Public domain W3C validator