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Theorem bj-dfmpoa 35289
Description: An equivalent definition of df-mpo 7280. (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 7280 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)}
2 dfoprab2 7333 . 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 623 . . . . . . . 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 644 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
74, 5, 63bitr2i 299 . . . . . . 7 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
87exbii 1850 . . . . . 6 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9 df-rex 3070 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10 r19.42v 3279 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
118, 9, 103bitr2i 299 . . . . 5 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1211exbii 1850 . . . 4 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
13 df-rex 3070 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1412, 13bitr4i 277 . . 3 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
1514opabbii 5141 . 2 {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
161, 2, 153eqtri 2770 1 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wex 1782  wcel 2106  wrex 3065  cop 4567  {copab 5136  {coprab 7276  cmpo 7277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-oprab 7279  df-mpo 7280
This theorem is referenced by:  bj-mpomptALT  35290
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