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Theorem fvpr1 7202
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 7199 . 2 ((𝐴 ∈ V ∧ 𝐶 ∈ V ∧ 𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
41, 2, 3mp3an12 1448 1 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wne 2937  Vcvv 3471  {cpr 4631  cop 4635  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556
This theorem is referenced by:  fvpr2OLD  7205  fprb  7206  fvtp1  7207  fnprb  7220  m2detleiblem3  22530  m2detleiblem4  22531  axlowdimlem6  28757  umgr2v2evd2  29340  bj-endbase  36795  poimirlem22  37115  nnsum3primes4  47128  nnsum3primesgbe  47132  zlmodzxzldeplem3  47570  zlmodzxzldeplem4  47571  2arymaptfo  47727  prelrrx2b  47787  rrx2plordisom  47796  ehl2eudisval0  47798  line2  47825  itscnhlinecirc02p  47858
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