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Theorem fvtp3 6345
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 4227 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}
21fveq1i 6089 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶)
3 necom 2834 . . . 4 (𝐴𝐶𝐶𝐴)
4 fvtp3.1 . . . . 5 𝐶 ∈ V
5 fvtp3.4 . . . . 5 𝐹 ∈ V
64, 5fvtp2 6344 . . . 4 ((𝐵𝐶𝐶𝐴) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
73, 6sylan2b 490 . . 3 ((𝐵𝐶𝐴𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
87ancoms 467 . 2 ((𝐴𝐶𝐵𝐶) → ({⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩, ⟨𝐴, 𝐷⟩}‘𝐶) = 𝐹)
92, 8syl5eq 2655 1 ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  {ctp 4128  cop 4130  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  fntpb  6356  wlkntrllem2  25856  constr3lem5  25942  rabren3dioph  36193  nnsum4primesodd  40010  nnsum4primesoddALTV  40011
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