Theorem bj-dfmpoa 37741

Description: An equivalent definition of df-mpo 7417. (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 7417	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)}
2		dfoprab2 7470	. 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 465	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶) ↔ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	anbi2i 634	. . . . . . . 8 ⊢ ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))))
5		anass 473	. . . . . . . 8 ⊢ (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))))
6		an13 659	. . . . . . . 8 ⊢ (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
7	4, 5, 6	3bitr2i 302	. . . . . . 7 ⊢ ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
8	7	exbii 1878	. . . . . 6 ⊢ (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9		df-rex 3090	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10		r19.42v 3197	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
11	8, 9, 10	3bitr2i 302	. . . . 5 ⊢ (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
12	11	exbii 1878	. . . 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 3090	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
14	12, 13	bitr4i 281	. . 3 ⊢ (∃𝑥∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
15	14	opabbii 5179	. 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 2790	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  ⟨cop 4596  {copab 5174  {coprab 7413   ∈ cmpo 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5175  df-oprab 7416  df-mpo 7417

This theorem is referenced by:  bj-mpomptALT  37742