Theorem tpf 14502

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 4722	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
2	1	3ad2ant1 1142	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
3		tpid2g 4724	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
4	3	3ad2ant2 1143	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
5		tpid3g 4725	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
6	5	3ad2ant3 1144	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
7	4, 6	ifcld 4521	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 1, 𝐵, 𝐶) ∈ {𝐴, 𝐵, 𝐶})
8	2, 7	ifcld 4521	. . . 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 2867	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)) ∈ 𝑇)
11	10	adantr 483	. 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 7084	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)

Syntax hints:   → wi 4   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ifcif 4474  {ctp 4580   ↦ cmpt 5175  ⟶wf 6506  (class class class)co 7385  0cc0 11063  1c1 11064  3c3 12263  ..^cfzo 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-fun 6512  df-fn 6513  df-f 6514

This theorem is referenced by:  tpfo  14503