Theorem mpofvd 35794

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

Hypotheses

Ref	Expression
mpofvd.1	⊢ (𝜑 → 𝑋 ∈ (𝐴 × 𝐵))
mpofvd.2	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mpofvd.3	⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦⟩) → 𝑌 = 𝐶)
mpofvd.4	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpofvd	⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

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

Proof of Theorem mpofvd

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpofvd.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐴 × 𝐵))
2		elxpi 5677	. . . 4 ⊢ (𝑋 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4		idd 25	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
5		mpofvd.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦⟩) → 𝑌 = 𝐶)
6	4, 5	jctird 536	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶)))
7	6	imdistanda 582	. . . 4 ⊢ (𝜑 → ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶))))
8	7	2eximdv 1954	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶))))
9	3, 8	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶)))
10	1	elexd 3477	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
11		mpofvd.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		eqeq1 2767	. . . . . 6 ⊢ (𝑤 = 𝑋 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 𝑦⟩))
13	12	anbi1d 643	. . . . 5 ⊢ (𝑤 = 𝑋 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ (𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
14	13	2exbidv 1959	. . . 4 ⊢ (𝑤 = 𝑋 → (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
15		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = 𝑌 → (𝑧 = 𝐶 ↔ 𝑌 = 𝐶))
16	15	anbi2d 642	. . . . . 6 ⊢ (𝑧 = 𝑌 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶)))
17	16	anbi2d 642	. . . . 5 ⊢ (𝑧 = 𝑌 → ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ (𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶))))
18	17	2exbidv 1959	. . . 4 ⊢ (𝑧 = 𝑌 → (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶))))
19		moeq 3668	. . . . . . 7 ⊢ ∃*𝑧 𝑧 = 𝐶
20	19	moani 2581	. . . . . 6 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
21	20	mosubop 5487	. . . . 5 ⊢ ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))
22	21	a1i 11	. . . 4 ⊢ (𝑤 ∈ V → ∃*𝑧∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
23		mpofvd.4	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
24		df-mpo 7445	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
25		dfoprab2 7498	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
26	23, 24, 25	3eqtri 2790	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))}
27		vex 3458	. . . . . . 7 ⊢ 𝑤 ∈ V
28	27	biantrur 540	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)) ↔ (𝑤 ∈ V ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))))
29	28	opabbii 5175	. . . . 5 ⊢ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ V ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
30	26, 29	eqtri 2786	. . . 4 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ V ∧ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))}
31	14, 18, 22, 30	fvopab3ig 7009	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝑉) → (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶)) → (𝐹‘𝑋) = 𝑌))
32	10, 11, 31	syl2anc 596	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑌 = 𝐶)) → (𝐹‘𝑋) = 𝑌))
33	9, 32	mpd 16	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃*wmo 2565  Vcvv 3454  ⟨cop 4838  {copab 5170   × cxp 5653  ‘cfv 6555  {coprab 7441   ∈ 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:  mpofvopd  35795