Theorem mpt3fvot2d 7688

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

Hypotheses

Ref	Expression
mpt3fvot2d.1	⊢ (𝜑 → 𝑅 ∈ 𝐴)
mpt3fvot2d.2	⊢ (𝜑 → 𝑆 ∈ 𝐵)
mpt3fvot2d.3	⊢ (𝜑 → 𝑇 ∈ 𝐶)
mpt3fvot2d.4	⊢ (𝜑 → 𝑀 ∈ 𝑉)
mpt3fvot2d.5	⊢ ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐷)
mpt3fvot2d.6	⊢ ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)
mpt3fvot2d.7	⊢ ((𝜑 ∧ 𝑇 = 𝑧) → 𝑀 = 𝐿)
mpt3fvot2d.8	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)

Assertion

Ref	Expression
mpt3fvot2d	⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑀)

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,𝑀,𝑦,𝑧

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

Proof of Theorem mpt3fvot2d

Step	Hyp	Ref	Expression
1		mpt3fvot2d.1	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
2		mpt3fvot2d.2	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
3		mpt3fvot2d.3	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝐶)
4		mpt3fvot2d.4	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑉)
5		otthg 5454	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐶) → (⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ↔ (𝑅 = 𝑥 ∧ 𝑆 = 𝑦 ∧ 𝑇 = 𝑧)))
6	1, 2, 3, 5	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ↔ (𝑅 = 𝑥 ∧ 𝑆 = 𝑦 ∧ 𝑇 = 𝑧)))
7		mpt3fvot2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐷)
8	7	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑅 = 𝑥 → 𝐾 = 𝐷))
9		mpt3fvot2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)
10	9	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑆 = 𝑦 → 𝐿 = 𝐾))
11		mpt3fvot2d.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 𝑧) → 𝑀 = 𝐿)
12	11	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑇 = 𝑧 → 𝑀 = 𝐿))
13	8, 10, 12	3anim123d 1471	. . . . 5 ⊢ (𝜑 → ((𝑅 = 𝑥 ∧ 𝑆 = 𝑦 ∧ 𝑇 = 𝑧) → (𝐾 = 𝐷 ∧ 𝐿 = 𝐾 ∧ 𝑀 = 𝐿)))
14	6, 13	sylbid 243	. . . 4 ⊢ (𝜑 → (⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → (𝐾 = 𝐷 ∧ 𝐿 = 𝐾 ∧ 𝑀 = 𝐿)))
15		eqtr 2781	. . . . . . 7 ⊢ ((𝐿 = 𝐾 ∧ 𝐾 = 𝐷) → 𝐿 = 𝐷)
16		eqtr 2781	. . . . . . . 8 ⊢ ((𝑀 = 𝐿 ∧ 𝐿 = 𝐷) → 𝑀 = 𝐷)
17	16	ancoms 464	. . . . . . 7 ⊢ ((𝐿 = 𝐷 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐷)
18	15, 17	sylan 592	. . . . . 6 ⊢ (((𝐿 = 𝐾 ∧ 𝐾 = 𝐷) ∧ 𝑀 = 𝐿) → 𝑀 = 𝐷)
19	18	ancom1s 666	. . . . 5 ⊢ (((𝐾 = 𝐷 ∧ 𝐿 = 𝐾) ∧ 𝑀 = 𝐿) → 𝑀 = 𝐷)
20	19	3impa 1127	. . . 4 ⊢ ((𝐾 = 𝐷 ∧ 𝐿 = 𝐾 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐷)
21	14, 20	syl6 36	. . 3 ⊢ (𝜑 → (⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑀 = 𝐷))
22	21	imp 412	. 2 ⊢ ((𝜑 ∧ ⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑀 = 𝐷)
23		mpt3fvot2d.8	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
24	1, 2, 3, 4, 22, 23	mpt3fvotd 7687	1 ⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cotp 4592  ‘cfv 6537   ∈ cmpt3 7681

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-mpt3 7682

This theorem is used by: (None)