Theorem mpt3fvd 7686

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

Hypotheses

Ref	Expression
mpt3fvd.1	⊢ (𝜑 → 𝑋 ∈ ((𝐴 × 𝐵) × 𝐶))
mpt3fvd.2	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mpt3fvd.3	⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)
mpt3fvd.4	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)

Assertion

Ref	Expression
mpt3fvd	⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

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

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

Proof of Theorem mpt3fvd

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

Step	Hyp	Ref	Expression
1		mpt3fvd.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((𝐴 × 𝐵) × 𝐶))
2		el2xptp 5820	. . . 4 ⊢ (𝑋 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩)
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩)
4		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩)
5		mpt3fvd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)
6	4, 5	jca 521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩) → (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷))
7	6	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
8	7	reximdv 3178	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ 𝐶 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
9	8	reximdv 3178	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
10	9	reximdv 3178	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
11	3, 10	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷))
12	1	elexd 3474	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
13		mpt3fvd.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14		eqeq1 2765	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ↔ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩))
15	14	anbi1d 643	. . . . . . 7 ⊢ (𝑣 = 𝑋 → ((𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
16	15	rexbidv 3187	. . . . . 6 ⊢ (𝑣 = 𝑋 → (∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
17	16	2rexbidv 3228	. . . . 5 ⊢ (𝑣 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
18		eqeq1 2765	. . . . . . . 8 ⊢ (𝑤 = 𝑌 → (𝑤 = 𝐷 ↔ 𝑌 = 𝐷))
19	18	anbi2d 642	. . . . . . 7 ⊢ (𝑤 = 𝑌 → ((𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
20	19	rexbidv 3187	. . . . . 6 ⊢ (𝑤 = 𝑌 → (∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
21	20	2rexbidv 3228	. . . . 5 ⊢ (𝑤 = 𝑌 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷)))
22		moeq 3665	. . . . . . . . . . 11 ⊢ ∃*𝑤 𝑤 = 𝐷
23	22	moani 2579	. . . . . . . . . 10 ⊢ ∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
24	23	ax-gen 1828	. . . . . . . . 9 ⊢ ∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
25	24	gen2 1829	. . . . . . . 8 ⊢ ∀𝑥∀𝑦∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
26		mosubott 5484	. . . . . . . 8 ⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷) → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷))
28		r3ex 3202	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
29		an12 658	. . . . . . . . . 10 ⊢ ((𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
30	29	3exbii 1883	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
31	28, 30	bitr4i 281	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
32	31	mobii 2574	. . . . . . 7 ⊢ (∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
33	27, 32	mpbir 234	. . . . . 6 ⊢ ∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)
34	33	a1i 11	. . . . 5 ⊢ (𝑣 ∈ V → ∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷))
35		df-mpt3 7682	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)}
36		vex 3455	. . . . . . . 8 ⊢ 𝑣 ∈ V
37	36	biantrur 540	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ (𝑣 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
38	37	opabbii 5172	. . . . . 6 ⊢ {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)} = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷))}
39	35, 38	eqtri 2784	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑣, 𝑤⟩ ∣ (𝑣 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷))}
40	17, 21, 34, 39	fvopab3ig 6987	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)‘𝑋) = 𝑌))
41		mpt3fvd.4	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
42	41	fveq1i 6884	. . . . 5 ⊢ (𝐹‘𝑋) = ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)‘𝑋)
43	42	eqeq1i 2766	. . . 4 ⊢ ((𝐹‘𝑋) = 𝑌 ↔ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)‘𝑋) = 𝑌)
44	40, 43	imbitrrdi 255	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷) → (𝐹‘𝑋) = 𝑌))
45	12, 13, 44	syl2anc 596	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑋 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑌 = 𝐷) → (𝐹‘𝑋) = 𝑌))
46	11, 45	mpd 16	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃wrex 3087  Vcvv 3451  ⟨cotp 4592  {copab 5167   × cxp 5649  ‘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:  mpt3fvotd  7687