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Theorem fvtp1 6602
Description: The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp1.1 𝐴 ∈ V
fvtp1.4 𝐷 ∈ V
Assertion
Ref Expression
fvtp1 ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)

Proof of Theorem fvtp1
StepHypRef Expression
1 df-tp 4318 . . 3 {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})
21fveq1i 6332 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴)
3 necom 2994 . . . 4 (𝐴𝐶𝐶𝐴)
4 fvunsn 6587 . . . 4 (𝐶𝐴 → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴))
53, 4sylbi 207 . . 3 (𝐴𝐶 → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴))
6 fvtp1.1 . . . 4 𝐴 ∈ V
7 fvtp1.4 . . . 4 𝐷 ∈ V
86, 7fvpr1 6598 . . 3 (𝐴𝐵 → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩}‘𝐴) = 𝐷)
95, 8sylan9eqr 2825 . 2 ((𝐴𝐵𝐴𝐶) → (({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩} ∪ {⟨𝐶, 𝐹⟩})‘𝐴) = 𝐷)
102, 9syl5eq 2815 1 ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wcel 2143  wne 2941  Vcvv 3348  cun 3718  {csn 4313  {cpr 4315  {ctp 4317  cop 4319  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-res 5260  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by:  fvtp2  6603  fntpb  6615  rabren3dioph  37905  nnsum4primesodd  42209  nnsum4primesoddALTV  42210
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