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Theorem elidinxp 5784
 Description: Characterization of elements of the intersection of identity relation with 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 3228 . . . . 5 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑦 = 𝑥)
21anbi2ci 624 . . . 4 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
3 r19.42v 3308 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦𝐵 𝑦 = 𝑥))
4 opeq2 4705 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
54equcoms 2002 . . . . . . . 8 (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
65eqeq2d 2803 . . . . . . 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 3435 . . . . . . . . 9 𝑦 ∈ V
98ideq 5601 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
10 df-br 4957 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11 equcom 2000 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
129, 10, 113bitr3i 302 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
1312anbi2i 622 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
147, 13bitr4i 279 . . . . 5 ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1514rexbii 3209 . . . 4 (∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
162, 3, 153bitr2i 300 . . 3 ((𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1716rexbii 3209 . 2 (∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18 rexin 4131 . 2 (∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 (𝑥𝐵𝐶 = ⟨𝑥, 𝑥⟩))
19 elinxp 5763 . 2 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝐴𝑦𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2017, 18, 193bitr4ri 305 1 (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴𝐵)𝐶 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1520   ∈ wcel 2079  ∃wrex 3104   ∩ cin 3853  ⟨cop 4472   class class class wbr 4956   I cid 5339   × cxp 5433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019  df-id 5340  df-xp 5441  df-rel 5442 This theorem is referenced by:  elidinxpid  5785  elrid  5786
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