Theorem eqer 8780

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 5833	. 2 ⊢ Rel 𝑅
3		id 22	. . . 4 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
4	3	eqcomd 2741	. . 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 8779	. . 3 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
7	5, 1	eqerlem 8779	. . 3 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	4, 6, 7	3imtr4i 292	. 2 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧)
9		eqtr 2758	. . 3 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
10	5, 1	eqerlem 8779	. . . 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 8779	. . 3 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
13	9, 11, 12	3imtr4i 292	. 2 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣)
14		vex 3482	. . 3 ⊢ 𝑧 ∈ V
15		eqid 2735	. . . 4 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
16	5, 1	eqerlem 8779	. . . 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 8771	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⦋csb 3908   class class class wbr 5148  {copab 5210   Er wer 8741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-er 8744

This theorem is referenced by:  ider  8781  frgpuplem  19805  fneer  36336