Theorem eqer 6619

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.)

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	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
2	1	relopabi 4787	. . . 4 ⊢ Rel 𝑅
3	2	a1i 9	. . 3 ⊢ (⊤ → Rel 𝑅)
4		id 19	. . . . . 6 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
5	4	eqcomd 2199	. . . . 5 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
6		eqer.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
7	6, 1	eqerlem 6618	. . . . 5 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
8	6, 1	eqerlem 6618	. . . . 5 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9	5, 7, 8	3imtr4i 201	. . . 4 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧)
10	9	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11		eqtr 2211	. . . . 5 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
12	6, 1	eqerlem 6618	. . . . . 6 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
13	7, 12	anbi12i 460	. . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴))
14	6, 1	eqerlem 6618	. . . . 5 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
15	11, 13, 14	3imtr4i 201	. . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣)
16	15	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17		vex 2763	. . . . 5 ⊢ 𝑧 ∈ V
18		eqid 2193	. . . . . 6 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
19	6, 1	eqerlem 6618	. . . . . 6 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
20	18, 19	mpbir 146	. . . . 5 ⊢ 𝑧𝑅𝑧
21	17, 20	2th 174	. . . 4 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
22	21	a1i 9	. . 3 ⊢ (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
23	3, 10, 16, 22	iserd 6613	. 2 ⊢ (⊤ → 𝑅 Er V)
24	23	mptru 1373	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  Vcvv 2760  ⦋csb 3080   class class class wbr 4029  {copab 4089  Rel wrel 4664   Er wer 6584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-er 6587

This theorem is referenced by:  ider  6620