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Theorem elidinxp 5884
 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 3192 . . . . 5 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑥)
21anbi2ci 628 . . . 4 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
3 r19.42v 3269 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
4 opeq2 4764 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
54equcoms 2028 . . . . . . . 8 (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
65eqeq2d 2770 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
76pm5.32ri 580 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8 vex 3414 . . . . . . . . 9 𝑦 ∈ V
98ideq 5693 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
10 df-br 5034 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11 equcom 2026 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
129, 10, 113bitr3i 305 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
1312anbi2i 626 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
147, 13bitr4i 281 . . . . 5 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1514rexbii 3176 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
162, 3, 153bitr2i 303 . . 3 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1716rexbii 3176 . 2 (∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 rexin 4145 . 2 (∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩))
19 elinxp 5862 . 2 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2017, 18, 193bitr4ri 308 1 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072   ∩ cin 3858  ⟨cop 4529   class class class wbr 5033   I cid 5430   × cxp 5523 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532 This theorem is referenced by:  elidinxpid  5885  elrid  5886  idinxpres  5887
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