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Theorem tpf 14535
Description: A function into a (proper) triple. (Contributed by AV, 20-Jul-2025.)
Hypotheses
Ref Expression
tpf1o.f 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))
tpf.t 𝑇 = {𝐴, 𝐵, 𝐶}
Assertion
Ref Expression
tpf ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐹:(0..^3)⟶𝑇)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝑥,𝐵   𝑥,𝐶   𝑥,𝑇
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem tpf
StepHypRef Expression
1 tpid1g 4774 . . . . . 6 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵, 𝐶})
213ad2ant1 1132 . . . . 5 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3 tpid2g 4776 . . . . . . 7 (𝐵𝑉𝐵 ∈ {𝐴, 𝐵, 𝐶})
433ad2ant2 1133 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
5 tpid3g 4777 . . . . . . 7 (𝐶𝑉𝐶 ∈ {𝐴, 𝐵, 𝐶})
653ad2ant3 1134 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
74, 6ifcld 4577 . . . . 5 ((𝐴𝑉𝐵𝑉𝐶𝑉) → if(𝑥 = 1, 𝐵, 𝐶) ∈ {𝐴, 𝐵, 𝐶})
82, 7ifcld 4577 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ {𝐴, 𝐵, 𝐶})
9 tpf.t . . . 4 𝑇 = {𝐴, 𝐵, 𝐶}
108, 9eleqtrrdi 2850 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ 𝑇)
1110adantr 480 . 2 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝑥 ∈ (0..^3)) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ 𝑇)
12 tpf1o.f . 2 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))
1311, 12fmptd 7134 1 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐹:(0..^3)⟶𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  ifcif 4531  {ctp 4635  cmpt 5231  wf 6559  (class class class)co 7431  0cc0 11153  1c1 11154  3c3 12320  ..^cfzo 13691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  tpfo  14536
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