Theorem tpf 14526

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 4731	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
2	1	3ad2ant1 1149	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		tpid2g 4733	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
4	3	3ad2ant2 1150	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
5		tpid3g 4734	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6	5	3ad2ant3 1151	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
7	4, 6	ifcld 4530	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 1, 𝐵, 𝐶) ∈ {𝐴, 𝐵, 𝐶})
8	2, 7	ifcld 4530	. . . 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 2876	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ 𝑇)
11	10	adantr 485	. 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 7099	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ifcif 4483  {ctp 4589   ↦ cmpt 5186  ⟶wf 6521  (class class class)co 7400  0cc0 11088  1c1 11089  3c3 12287  ..^cfzo 13673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-fun 6527  df-fn 6528  df-f 6529

This theorem is referenced by:  tpfo  14527