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Theorem elidinxp 6043
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 3230 . . . . 5 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑥)
21anbi2ci 625 . . . 4 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
3 r19.42v 3190 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
4 opeq2 4874 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
54equcoms 2023 . . . . . . . 8 (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
65eqeq2d 2743 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
76pm5.32ri 576 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8 vex 3478 . . . . . . . . 9 𝑦 ∈ V
98ideq 5852 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
10 df-br 5149 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11 equcom 2021 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
129, 10, 113bitr3i 300 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
1312anbi2i 623 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
147, 13bitr4i 277 . . . . 5 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1514rexbii 3094 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
162, 3, 153bitr2i 298 . . 3 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1716rexbii 3094 . 2 (∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 rexin 4239 . 2 (∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩))
19 elinxp 6019 . 2 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2017, 18, 193bitr4ri 303 1 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  cin 3947  cop 4634   class class class wbr 5148   I cid 5573   × cxp 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683
This theorem is referenced by:  elidinxpid  6044  elrid  6045  idinxpres  6046
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