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Theorem eqer 7774
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 = 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 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21relopabi 5243 . 2 Rel 𝑅
3 id 22 . . . 4 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
43eqcomd 2627 . . 3 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
5 eqer.1 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
65, 1eqerlem 7773 . . 3 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
75, 1eqerlem 7773 . . 3 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
84, 6, 73imtr4i 281 . 2 (𝑧𝑅𝑤𝑤𝑅𝑧)
9 eqtr 2640 . . 3 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
105, 1eqerlem 7773 . . . 4 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
116, 10anbi12i 733 . . 3 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
125, 1eqerlem 7773 . . 3 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
139, 11, 123imtr4i 281 . 2 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
14 vex 3201 . . 3 𝑧 ∈ V
15 eqid 2621 . . . 4 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
165, 1eqerlem 7773 . . . 4 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
1715, 16mpbir 221 . . 3 𝑧𝑅𝑧
1814, 172th 254 . 2 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
192, 8, 13, 18iseri 7766 1 𝑅 Er V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  csb 3531   class class class wbr 4651  {copab 4710   Er wer 7736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-er 7739
This theorem is referenced by:  ider  7776  frgpuplem  18179  fneer  32332
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