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Theorem fvpr1 7140
Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by BJ, 26-Sep-2024.)
Hypotheses
Ref Expression
fvpr1.1 𝐴 ∈ V
fvpr1.2 𝐶 ∈ V
Assertion
Ref Expression
fvpr1 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Proof of Theorem fvpr1
StepHypRef Expression
1 fvpr1.1 . 2 𝐴 ∈ V
2 fvpr1.2 . 2 𝐶 ∈ V
3 fvpr1g 7137 . 2 ((𝐴 ∈ V ∧ 𝐶 ∈ V ∧ 𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
41, 2, 3mp3an12 1452 1 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2940  Vcvv 3444  {cpr 4589  cop 4593  cfv 6497
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-10 2138  ax-11 2155  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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  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-uni 4867  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-res 5646  df-iota 6449  df-fun 6499  df-fv 6505
This theorem is referenced by:  fvpr2OLD  7143  fprb  7144  fvtp1  7145  fnprb  7159  m2detleiblem3  21994  m2detleiblem4  21995  axlowdimlem6  27938  umgr2v2evd2  28517  bj-endbase  35833  poimirlem22  36146  nnsum3primes4  46066  nnsum3primesgbe  46070  zlmodzxzldeplem3  46669  zlmodzxzldeplem4  46670  2arymaptfo  46826  prelrrx2b  46886  rrx2plordisom  46895  ehl2eudisval0  46897  line2  46924  itscnhlinecirc02p  46957
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