Theorem bj-dfmpoa 37078

Description: An equivalent definition of df-mpo 7418. (Contributed by BJ, 30-Dec-2020.)

Assertion

Ref	Expression
bj-dfmpoa	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Distinct variable groups:   𝑥,𝑦,𝑠,𝑡   𝐴,𝑠,𝑡   𝐵,𝑠,𝑡   𝐶,𝑠,𝑡   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem bj-dfmpoa

Step	Hyp	Ref	Expression
1		df-mpo 7418	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)}
2		dfoprab2 7473	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶))}
3		ancom 460	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶) ↔ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	anbi2i 623	. . . . . . . 8 ⊢ ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))))
5		anass 468	. . . . . . . 8 ⊢ (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))))
6		an13 647	. . . . . . . 8 ⊢ (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
7	4, 5, 6	3bitr2i 299	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
8	7	exbii 1847	. . . . . 6 ⊢ (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9		df-rex 3060	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10		r19.42v 3178	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
11	8, 9, 10	3bitr2i 299	. . . . 5 ⊢ (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
12	11	exbii 1847	. . . 4 ⊢ (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
13		df-rex 3060	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
14	12, 13	bitr4i 278	. . 3 ⊢ (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
15	14	opabbii 5190	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ ∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶))} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
16	1, 2, 15	3eqtri 2761	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃wrex 3059  ⟨cop 4612  {copab 5185  {coprab 7414   ∈ cmpo 7415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5186  df-oprab 7417  df-mpo 7418

This theorem is referenced by:  bj-mpomptALT  37079