Theorem eqer 8658

Description: Equivalence relation involving equality of dependent classes 𝐴(𝑥) and 𝐵(𝑦). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof shortened by AV, 1-May-2021.)

Hypotheses

Ref	Expression
eqer.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
eqer.2	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}

Assertion

Ref	Expression
eqer	⊢ 𝑅 Er V

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem eqer

Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
2	1	relopabiv 5760	. 2 ⊢ Rel 𝑅
3		id 22	. . . 4 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
4	3	eqcomd 2737	. . 3 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
5		eqer.1	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
6	5, 1	eqerlem 8657	. . 3 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
7	5, 1	eqerlem 8657	. . 3 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	4, 6, 7	3imtr4i 292	. 2 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧)
9		eqtr 2751	. . 3 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
10	5, 1	eqerlem 8657	. . . 4 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
11	6, 10	anbi12i 628	. . 3 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴))
12	5, 1	eqerlem 8657	. . 3 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
13	9, 11, 12	3imtr4i 292	. 2 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣)
14		vex 3440	. . 3 ⊢ 𝑧 ∈ V
15		eqid 2731	. . . 4 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
16	5, 1	eqerlem 8657	. . . 4 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
17	15, 16	mpbir 231	. . 3 ⊢ 𝑧𝑅𝑧
18	14, 17	2th 264	. 2 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
19	2, 8, 13, 18	iseri 8649	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3850   class class class wbr 5091  {copab 5153   Er wer 8619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-er 8622

This theorem is referenced by:  ider  8659  frgpuplem  19682  fneer  36386