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Theorem elidinxp 6064
Description: Characterization of the elements of the intersection of the identity relation with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elidinxp (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elidinxp
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 risset 3231 . . . . 5 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑥)
21anbi2ci 625 . . . 4 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
3 r19.42v 3189 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
4 opeq2 4879 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
54equcoms 2017 . . . . . . . 8 (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
65eqeq2d 2746 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
76pm5.32ri 575 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8 vex 3482 . . . . . . . . 9 𝑦 ∈ V
98ideq 5866 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
10 df-br 5149 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11 equcom 2015 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
129, 10, 113bitr3i 301 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
1312anbi2i 623 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
147, 13bitr4i 278 . . . . 5 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1514rexbii 3092 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
162, 3, 153bitr2i 299 . . 3 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1716rexbii 3092 . 2 (∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 rexin 4256 . 2 (∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩))
19 elinxp 6039 . 2 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2017, 18, 193bitr4ri 304 1 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cin 3962  cop 4637   class class class wbr 5148   I cid 5582   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696
This theorem is referenced by:  elidinxpid  6065  elrid  6066  idinxpres  6067
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