Theorem mpt3mpt 7685

Description: Express a three-argument function as a one-argument function, or vice-versa. (Contributed by BTernaryTau, 7-Sep-2026.)

Hypothesis

Ref	Expression
mpt3mpt.1	⊢ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝐷 = 𝐸)

Assertion

Ref	Expression
mpt3mpt	⊢ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ↦ 𝐷) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐸)

Distinct variable groups:   𝑤,𝐸   𝑥,𝐷,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐷(𝑤)   𝐸(𝑥, 𝑦, 𝑧)

Proof of Theorem mpt3mpt

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el2xptp 5819	. . . . 5 ⊢ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩)
2	1	anbi1i 636	. . . 4 ⊢ ((𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑡 = 𝐷) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷))
3		r19.41v 3192	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷))
4		r19.41v 3192	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷))
5		r19.41v 3192	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ (∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷))
6		mpt3mpt.1	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝐷 = 𝐸)
7	6	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝑡 = 𝐷 ↔ 𝑡 = 𝐸))
8	7	pm5.32i 585	. . . . . . . . 9 ⊢ ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
9	8	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
10	5, 9	bitr3i 280	. . . . . . 7 ⊢ ((∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
11	10	rexbii 3109	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
12	4, 11	bitr3i 280	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
13	12	rexbii 3109	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
14	2, 3, 13	3bitr2i 302	. . 3 ⊢ ((𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑡 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸))
15	14	opabbii 5171	. 2 ⊢ {⟨𝑤, 𝑡⟩ ∣ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑡 = 𝐷)} = {⟨𝑤, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸)}
16		df-mpt 5186	. 2 ⊢ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ↦ 𝐷) = {⟨𝑤, 𝑡⟩ ∣ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑡 = 𝐷)}
17		df-mpt3 7684	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐸) = {⟨𝑤, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐸)}
18	15, 16, 17	3eqtr4i 2793	1 ⊢ (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ↦ 𝐷) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐸)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ⟨cotp 4591  {copab 5166   ↦ cmpt 5185   × cxp 5648   ∈ cmpt3 7683

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-iun 4952  df-opab 5167  df-mpt 5186  df-xp 5656  df-rel 5657  df-mpt3 7684

This theorem is used by: (None)