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Theorem fntp 6563
Description: A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
fntp.1 𝐴 ∈ V
fntp.2 𝐵 ∈ V
fntp.3 𝐶 ∈ V
fntp.4 𝐷 ∈ V
fntp.5 𝐸 ∈ V
fntp.6 𝐹 ∈ V
Assertion
Ref Expression
fntp ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})

Proof of Theorem fntp
StepHypRef Expression
1 fntp.1 . . 3 𝐴 ∈ V
2 fntp.2 . . 3 𝐵 ∈ V
3 fntp.3 . . 3 𝐶 ∈ V
4 fntp.4 . . 3 𝐷 ∈ V
5 fntp.5 . . 3 𝐸 ∈ V
6 fntp.6 . . 3 𝐹 ∈ V
71, 2, 3, 4, 5, 6funtp 6559 . 2 ((𝐴𝐵𝐴𝐶𝐵𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
84, 5, 6dmtpop 6171 . 2 dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}
9 df-fn 6500 . 2 ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶} ↔ (Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ∧ dom {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} = {𝐴, 𝐵, 𝐶}))
107, 8, 9sylanblrc 591 1 ((𝐴𝐵𝐴𝐶𝐵𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2940  Vcvv 3444  {ctp 4591  cop 4593  dom cdm 5634  Fun wfun 6491   Fn wfn 6492
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-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-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  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-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  fntpb  7160  rabren3dioph  41181
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