Theorem mpofvop2d 35796

Description: Value of a two-argument function (usually an operation) in maps-to notation at an ordered pair. (Contributed by BTernaryTau, 18-Oct-2026.)

Hypotheses

Ref	Expression
mpofvop2d.1	⊢ (𝜑 → 𝑅 ∈ 𝐴)
mpofvop2d.2	⊢ (𝜑 → 𝑆 ∈ 𝐵)
mpofvop2d.3	⊢ (𝜑 → 𝐿 ∈ 𝑉)
mpofvop2d.4	⊢ ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐶)
mpofvop2d.5	⊢ ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)
mpofvop2d.6	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpofvop2d	⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆⟩) = 𝐿)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐿,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐾(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem mpofvop2d

Step	Hyp	Ref	Expression
1		mpofvop2d.1	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
2		mpofvop2d.2	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
3		mpofvop2d.3	. 2 ⊢ (𝜑 → 𝐿 ∈ 𝑉)
4		opthg 5450	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑅 = 𝑥 ∧ 𝑆 = 𝑦)))
5	1, 2, 4	syl2anc 596	. . . . 5 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑅 = 𝑥 ∧ 𝑆 = 𝑦)))
6		mpofvop2d.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐶)
7	6	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑅 = 𝑥 → 𝐾 = 𝐶))
8		mpofvop2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)
9	8	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑆 = 𝑦 → 𝐿 = 𝐾))
10	7, 9	anim12d 621	. . . . 5 ⊢ (𝜑 → ((𝑅 = 𝑥 ∧ 𝑆 = 𝑦) → (𝐾 = 𝐶 ∧ 𝐿 = 𝐾)))
11	5, 10	sylbid 243	. . . 4 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩ → (𝐾 = 𝐶 ∧ 𝐿 = 𝐾)))
12		eqtr 2783	. . . . 5 ⊢ ((𝐿 = 𝐾 ∧ 𝐾 = 𝐶) → 𝐿 = 𝐶)
13	12	ancoms 464	. . . 4 ⊢ ((𝐾 = 𝐶 ∧ 𝐿 = 𝐾) → 𝐿 = 𝐶)
14	11, 13	syl6 36	. . 3 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩ → 𝐿 = 𝐶))
15	14	imp 412	. 2 ⊢ ((𝜑 ∧ ⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩) → 𝐿 = 𝐶)
16		mpofvop2d.6	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
17	1, 2, 3, 15, 16	mpofvopd 35795	1 ⊢ (𝜑 → (𝐹‘⟨𝑅, 𝑆⟩) = 𝐿)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  ‘cfv 6555   ∈ cmpo 7442

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 5252  ax-pr 5395

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-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 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-oprab 7444  df-mpo 7445

This theorem is used by:  gdlop1fv  35962  gdlop2fv  35963  gdlop3fv  35964  gdlop4fv  35965  gdlop5fv  35966  gdlop6fv  35967  gdlop7fv  35968  gdlop8fv  35969