Theorem eqer 6720

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 4847	. . . 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 2235	. . . . 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 6719	. . . . 5 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
8	6, 1	eqerlem 6719	. . . . 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 2247	. . . . 5 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
12	6, 1	eqerlem 6719	. . . . . 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 6719	. . . . 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 2802	. . . . 5 ⊢ 𝑧 ∈ V
18		eqid 2229	. . . . . 6 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
19	6, 1	eqerlem 6719	. . . . . 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 6714	. 2 ⊢ (⊤ → 𝑅 Er V)
24	23	mptru 1404	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  Vcvv 2799  ⦋csb 3124   class class class wbr 4083  {copab 4144  Rel wrel 4724   Er wer 6685

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-er 6688

This theorem is referenced by:  ider  6721