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Theorem bj-dfmpoa 34290
Description: An equivalent definition of df-mpo 7156. (Contributed by BJ, 30-Dec-2020.)
Assertion
Ref Expression
bj-dfmpoa (𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (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
StepHypRef Expression
1 df-mpo 7156 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)}
2 dfoprab2 7207 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))}
3 ancom 461 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶) ↔ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
43anbi2i 622 . . . . . . . 8 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
5 anass 469 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
6 an13 643 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
74, 5, 63bitr2i 300 . . . . . . 7 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
87exbii 1841 . . . . . 6 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9 df-rex 3148 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10 r19.42v 3354 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
118, 9, 103bitr2i 300 . . . . 5 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1211exbii 1841 . . . 4 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
13 df-rex 3148 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1412, 13bitr4i 279 . . 3 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
1514opabbii 5129 . 2 {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
161, 2, 153eqtri 2852 1 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wex 1773  wcel 2106  wrex 3143  cop 4569  {copab 5124  {coprab 7152  cmpo 7153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  bj-mpomptALT  34291
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