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Theorem bj-dfmpt2a 33375
Description: An equivalent definition of df-mpt2 6816. (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 6816 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)}
2 dfoprab2 6864 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))}
3 ancom 465 . . . . . . . . 9 (((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶) ↔ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
43anbi2i 732 . . . . . . . 8 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
5 anass 684 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ (𝑡 = 𝐶 ∧ (𝑥𝐴𝑦𝐵))))
6 an13 875 . . . . . . . 8 (((𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ∧ (𝑥𝐴𝑦𝐵)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
74, 5, 63bitr2i 288 . . . . . . 7 ((𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
87exbii 1921 . . . . . 6 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
9 df-rex 3054 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))))
10 r19.42v 3228 . . . . . 6 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
118, 9, 103bitr2i 288 . . . . 5 (∃𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1211exbii 1921 . . . 4 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
13 df-rex 3054 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)))
1412, 13bitr4i 267 . . 3 (∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶)) ↔ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
1514opabbii 4867 . 2 {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝑦(𝑠 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑡 = 𝐶))} = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
161, 2, 153eqtri 2784 1 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1630  wex 1851  wcel 2137  wrex 3049  cop 4325  {copab 4862  {coprab 6812  cmpt2 6813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-opab 4863  df-oprab 6815  df-mpt2 6816
This theorem is referenced by:  bj-mpt2mptALT  33376
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