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Theorem eqer 6461
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 = 𝑦𝐴 = 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.
StepHypRef Expression
1 eqer.2 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21relopabi 4665 . . . 4 Rel 𝑅
32a1i 9 . . 3 (⊤ → Rel 𝑅)
4 id 19 . . . . . 6 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
54eqcomd 2145 . . . . 5 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
6 eqer.1 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
76, 1eqerlem 6460 . . . . 5 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
86, 1eqerlem 6460 . . . . 5 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
95, 7, 83imtr4i 200 . . . 4 (𝑧𝑅𝑤𝑤𝑅𝑧)
109adantl 275 . . 3 ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11 eqtr 2157 . . . . 5 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
126, 1eqerlem 6460 . . . . . 6 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
137, 12anbi12i 455 . . . . 5 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
146, 1eqerlem 6460 . . . . 5 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
1511, 13, 143imtr4i 200 . . . 4 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
1615adantl 275 . . 3 ((⊤ ∧ (𝑧𝑅𝑤𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17 vex 2689 . . . . 5 𝑧 ∈ V
18 eqid 2139 . . . . . 6 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
196, 1eqerlem 6460 . . . . . 6 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2018, 19mpbir 145 . . . . 5 𝑧𝑅𝑧
2117, 202th 173 . . . 4 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
2221a1i 9 . . 3 (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
233, 10, 16, 22iserd 6455 . 2 (⊤ → 𝑅 Er V)
2423mptru 1340 1 𝑅 Er V
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wtru 1332  wcel 1480  Vcvv 2686  csb 3003   class class class wbr 3929  {copab 3988  Rel wrel 4544   Er wer 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-er 6429
This theorem is referenced by:  ider  6462
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