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Theorem fvtp3 7147
Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp3.1 𝐶 ∈ V
fvtp3.4 𝐹 ∈ V
Assertion
Ref Expression
fvtp3 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)

Proof of Theorem fvtp3
StepHypRef Expression
1 tprot 4711 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}
21fveq1i 6844 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶)
3 necom 2994 . . . 4 (𝐴𝐶𝐶𝐴)
4 fvtp3.1 . . . . 5 𝐶 ∈ V
5 fvtp3.4 . . . . 5 𝐹 ∈ V
64, 5fvtp2 7146 . . . 4 ((𝐵𝐶𝐶𝐴) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
73, 6sylan2b 595 . . 3 ((𝐵𝐶𝐴𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
87ancoms 460 . 2 ((𝐴𝐶𝐵𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
92, 8eqtrid 2785 1 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  Vcvv 3444  {ctp 4591  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-3or 1089  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-tp 4592  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:  fntpb  7160  rabren3dioph  41181  nnsum4primesodd  46074  nnsum4primesoddALTV  46075
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