Theorem bj-mpt2mptALT 32701

Description: Alternate proof of mpt2mpt 6706. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-mpt2mptALT.1	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)

Assertion

Ref	Expression
bj-mpt2mptALT	⊢ (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-mpt2mptALT

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp2 5097	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 730	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
3		r19.41v 3086	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
4		r19.41v 3086	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
5		bj-mpt2mptALT.1	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
6	5	eqeq2d 2636	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷))
7	6	pm5.32i 668	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
8	7	rexbii 3039	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
9	4, 8	bitr3i 266	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
10	9	rexbii 3039	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
11	2, 3, 10	3bitr2i 288	. . 3 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
12	11	opabbii 4684	. 2 ⊢ {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
13		df-mpt 4680	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)}
14		bj-dfmpt2a 32700	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
15	12, 13, 14	3eqtr4i 2658	1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∃wrex 2913  ⟨cop 4159  {copab 4677   ↦ cmpt 4678   × cxp 5077   ↦ cmpt2 6607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-opab 4679  df-mpt 4680  df-xp 5085  df-oprab 6609  df-mpt2 6610

This theorem is referenced by: (None)