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Theorem funpr 5932
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 471 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 funpr.3 . . 3 𝐶 ∈ V
5 funpr.4 . . 3 𝐷 ∈ V
64, 5pm3.2i 471 . 2 (𝐶 ∈ V ∧ 𝐷 ∈ V)
7 funprg 5928 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ 𝐴𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
83, 6, 7mp3an12 1412 1 (𝐴𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  wne 2791  Vcvv 3195  {cpr 4170  cop 4174  Fun wfun 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-fun 5878
This theorem is referenced by:  funtp  5933  fpr  6406  fnprb  6457  1sdom  8148
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