Theorem eqer 8710

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 5791	. 2 ⊢ Rel 𝑅
3		id 22	. . . 4 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
4	3	eqcomd 2767	. . 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 8709	. . 3 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
7	5, 1	eqerlem 8709	. . 3 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	4, 6, 7	3imtr4i 294	. 2 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧)
9		eqtr 2781	. . 3 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
10	5, 1	eqerlem 8709	. . . 4 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
11	6, 10	anbi12i 637	. . 3 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴))
12	5, 1	eqerlem 8709	. . 3 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
13	9, 11, 12	3imtr4i 294	. 2 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣)
14		vex 3457	. . 3 ⊢ 𝑧 ∈ V
15		eqid 2761	. . . 4 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
16	5, 1	eqerlem 8709	. . . 4 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
17	15, 16	mpbir 233	. . 3 ⊢ 𝑧𝑅𝑧
18	14, 17	2th 266	. 2 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
19	2, 8, 13, 18	iseri 8701	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⦋csb 3852   class class class wbr 5099  {copab 5161   Er wer 8670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-er 8673

This theorem is referenced by:  ider  8711  frgpuplem  19795  fneer  36677