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Theorem bj-dfmpt2a 33564
Description: An equivalent definition of df-mpt2 6883. (Contributed by BJ, 30-Dec-2020.)
Assertion
Ref Expression
bj-dfmpt2a (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-dfmpt2a
StepHypRef Expression
1 df-mpt2 6883 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)}
2 dfoprab2 6935 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))}
3 ancom 453 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶) ↔ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
43anbi2i 617 . . . . . . . 8 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
5 anass 461 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
6 an13 638 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
74, 5, 63bitr2i 291 . . . . . . 7 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
87exbii 1944 . . . . . 6 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9 df-rex 3095 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10 r19.42v 3273 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
118, 9, 103bitr2i 291 . . . . 5 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1211exbii 1944 . . . 4 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
13 df-rex 3095 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1412, 13bitr4i 270 . . 3 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
1514opabbii 4910 . 2 {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
161, 2, 153eqtri 2825 1 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wex 1875  wcel 2157  wrex 3090  cop 4374  {copab 4905  {coprab 6879  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  bj-mpt2mptALT  33565
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