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) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem elridOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elres 5672	. 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
2		opeq2 4625	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
3	2	eqeq2d 2836	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 = ⟨𝑥, 𝑥⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
4	3	pm5.32ri 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
7	6	ideq 5508	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
8	5, 7	bitr3i 269	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
9	8	anbi2i 618	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
10	4, 9	bitr4i 270	. . . . 5 ⊢ ((𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
11	10	exbii 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))
14	12, 13	mpbiran2 703	. . . 4 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑦) ↔ 𝐴 = ⟨𝑥, 𝑥⟩)
15	11, 14	bitr3i 269	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝐴 = ⟨𝑥, 𝑥⟩)
16	15	rexbii 3252	. 2 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
17	1, 16	bitri 267	1 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)

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)