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Theorem funpr 6622
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 470 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 funpr.3 . . 3 𝐶 ∈ V
5 funpr.4 . . 3 𝐷 ∈ V
64, 5pm3.2i 470 . 2 (𝐶 ∈ V ∧ 𝐷 ∈ V)
7 funprg 6620 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
83, 6, 7mp3an12 1453 1 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wne 2940  Vcvv 3480  {cpr 4628  cop 4632  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563
This theorem is referenced by:  funtp  6623  fpr  7174  fnprb  7228  1sdomOLD  9285
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