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Theorem elridOLD 5696
 Description: Obsolete proof of elrid 5695 as of 15-Sep-2022. (Contributed by BJ, 28-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elridOLD (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem elridOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elres 5672 . 2 (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2 opeq2 4625 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
32eqeq2d 2836 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 = ⟨𝑥, 𝑥⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
43pm5.32ri 573 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
5 df-br 4875 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
6 vex 3418 . . . . . . . . 9 𝑦 ∈ V
76ideq 5508 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
85, 7bitr3i 269 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
98anbi2i 618 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
104, 9bitr4i 270 . . . . 5 ((𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
1110exbii 1949 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
12 ax6evr 2121 . . . . 5 𝑦 𝑥 = 𝑦
13 19.42v 2054 . . . . 5 (∃𝑦(𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝐴 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 𝑥 = 𝑦))
1412, 13mpbiran2 703 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝐴 = ⟨𝑥, 𝑥⟩)
1511, 14bitr3i 269 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝐴 = ⟨𝑥, 𝑥⟩)
1615rexbii 3252 . 2 (∃𝑥𝑋𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
171, 16bitri 267 1 (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658  ∃wex 1880   ∈ wcel 2166  ∃wrex 3119  ⟨cop 4404   class class class wbr 4874   I cid 5250   ↾ cres 5345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-res 5355 This theorem is referenced by: (None)
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