Theorem mpt3fvotd 7721

Description: Value of a three-argument function in maps-to notation at an ordered triple. (Contributed by BTernaryTau, 28-Sep-2026.)

Hypotheses

Ref	Expression
mpt3fvotd.1	⊢ (𝜑 → 𝑅 ∈ 𝐴)
mpt3fvotd.2	⊢ (𝜑 → 𝑆 ∈ 𝐵)
mpt3fvotd.3	⊢ (𝜑 → 𝑇 ∈ 𝐶)
mpt3fvotd.4	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mpt3fvotd.5	⊢ ((𝜑 ∧ ⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)
mpt3fvotd.6	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)

Assertion

Ref	Expression
mpt3fvotd	⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑌)

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧

Allowed substitution hints:   𝐷(𝑥, 𝑦, 𝑧)   𝐹(𝑥, 𝑦, 𝑧)   𝑉(𝑥, 𝑦, 𝑧)

Proof of Theorem mpt3fvotd

Step	Hyp	Ref	Expression
1		mpt3fvotd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
2		mpt3fvotd.2	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
3		mpt3fvotd.3	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐶)
4		otelxp 5702	. . 3 ⊢ (⟨𝑅, 𝑆, 𝑇⟩ ∈ ((𝐴 × 𝐵) × 𝐶) ↔ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐶))
5	1, 2, 3, 4	syl3anbrc 1362	. 2 ⊢ (𝜑 → ⟨𝑅, 𝑆, 𝑇⟩ ∈ ((𝐴 × 𝐵) × 𝐶))
6		mpt3fvotd.4	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		mpt3fvotd.5	. 2 ⊢ ((𝜑 ∧ ⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)
8		mpt3fvotd.6	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
9	5, 6, 7, 8	mpt3fvd 7720	1 ⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cotp 4863   × cxp 5656  ‘cfv 6566   ∈ cmpt3 7715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-mpt3 7716

This theorem is used by:  mpt3fvot2d  7722