Theorem tpf 14459

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 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

Step	Hyp	Ref	Expression
1		tpid1g 4708	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
2	1	3ad2ant1 1139	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		tpid2g 4710	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
4	3	3ad2ant2 1140	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
5		tpid3g 4711	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6	5	3ad2ant3 1141	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
7	4, 6	ifcld 4508	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 1, 𝐵, 𝐶) ∈ {𝐴, 𝐵, 𝐶})
8	2, 7	ifcld 4508	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ {𝐴, 𝐵, 𝐶})
9		tpf.t	. . . 4 ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}
10	8, 9	eleqtrrdi 2851	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ 𝑇)
11	10	adantr 481	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 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)))
13	11, 12	fmptd 7062	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ifcif 4461  {ctp 4566   ↦ cmpt 5160  ⟶wf 6488  (class class class)co 7363  0cc0 11036  1c1 11037  3c3 12235  ..^cfzo 13606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  tpfo  14460