Theorem bj-mpomptALT 38008

Description: Alternate proof of mpompt 7526. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-mpomptALT.1	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)

Assertion

Ref	Expression
bj-mpomptALT	⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

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

Allowed substitution hints:   𝐶(𝑧)   𝐷(𝑥, 𝑦)

Proof of Theorem bj-mpomptALT

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp2 5675	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 636	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
3		r19.41v 3193	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
4		r19.41v 3193	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
5		bj-mpomptALT.1	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
6	5	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷))
7	6	pm5.32i 585	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
8	7	rexbii 3110	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
9	4, 8	bitr3i 280	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
10	9	rexbii 3110	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
11	2, 3, 10	3bitr2i 302	. . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
12	11	opabbii 5172	. 2 ⊢ {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
13		df-mpt 5187	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)}
14		bj-dfmpoa 38007	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
15	12, 13, 14	3eqtr4i 2794	1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  ⟨cop 4590  {copab 5167   ↦ cmpt 5186   × cxp 5649   ∈ cmpo 7414

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-11 2194  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-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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-opab 5168  df-mpt 5187  df-xp 5657  df-oprab 7416  df-mpo 7417

This theorem is used by: (None)