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Theorem elidinxp 6044
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 626 . . . 4 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
3 r19.42v 3191 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
4 opeq2 4875 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
54equcoms 2024 . . . . . . . 8 (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
65eqeq2d 2744 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
76pm5.32ri 577 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8 vex 3479 . . . . . . . . 9 𝑦 ∈ V
98ideq 5853 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
10 df-br 5150 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11 equcom 2022 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
129, 10, 113bitr3i 301 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
1312anbi2i 624 . . . . . 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 3095 . . . 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 3095 . 2 (∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 rexin 4240 . 2 (∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩))
19 elinxp 6020 . 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 205  wa 397   = wceq 1542  wcel 2107  wrex 3071  cin 3948  cop 4635   class class class wbr 5149   I cid 5574   × cxp 5675
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684
This theorem is referenced by:  elidinxpid  6045  elrid  6046  idinxpres  6047
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