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Theorem fvpr1 7212
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 7210 . 2 ((𝐴 ∈ V ∧ 𝐶 ∈ V ∧ 𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
41, 2, 3mp3an12 1453 1 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  {cpr 4628  cop 4632  cfv 6561
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-10 2141  ax-11 2157  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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  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-uni 4908  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-res 5697  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  fprb  7214  fvtp1  7215  fnprb  7228  m2detleiblem3  22635  m2detleiblem4  22636  axlowdimlem6  28962  umgr2v2evd2  29545  bj-endbase  37317  poimirlem22  37649  nnsum3primes4  47775  nnsum3primesgbe  47779  zlmodzxzldeplem3  48419  zlmodzxzldeplem4  48420  2arymaptfo  48575  prelrrx2b  48635  rrx2plordisom  48644  ehl2eudisval0  48646  line2  48673  itscnhlinecirc02p  48706
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