Theorem bj-mpomptALT 37573

Description: Alternate proof of mpompt 7506. (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 5669	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 633	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
3		r19.41v 3191	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
4		r19.41v 3191	. . . . . 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 582	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
8	7	rexbii 3108	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
9	4, 8	bitr3i 279	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
10	9	rexbii 3108	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
11	2, 3, 10	3bitr2i 301	. . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
12	11	opabbii 5166	. 2 ⊢ {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
13		df-mpt 5181	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)}
14		bj-dfmpoa 37572	. 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)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ⟨cop 4587  {copab 5161   ↦ cmpt 5180   × cxp 5643   ∈ cmpo 7394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-mpt 5181  df-xp 5651  df-oprab 7396  df-mpo 7397

This theorem is referenced by: (None)