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Theorem erth 7650
Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
erth.1 (𝜑𝑅 Er 𝑋)
erth.2 (𝜑𝐴𝑋)
Assertion
Ref Expression
erth (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem erth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl 471 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝜑)
2 erth.1 . . . . . . . . 9 (𝜑𝑅 Er 𝑋)
32ersymb 7615 . . . . . . . 8 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
43biimpa 499 . . . . . . 7 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51, 4jca 552 . . . . . 6 ((𝜑𝐴𝑅𝐵) → (𝜑𝐵𝑅𝐴))
62ertr 7616 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
76impl 647 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
85, 7sylan 486 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
92ertr 7616 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
109impl 647 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
118, 10impbida 872 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
12 vex 3170 . . . . 5 𝑥 ∈ V
13 erth.2 . . . . . 6 (𝜑𝐴𝑋)
1413adantr 479 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴𝑋)
15 elecg 7644 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑋) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1612, 14, 15sylancr 693 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
17 errel 7610 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
182, 17syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
19 brrelex2 5066 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
2018, 19sylan 486 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
21 elecg 7644 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2212, 20, 21sylancr 693 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2311, 16, 223bitr4d 298 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2423eqrdv 2602 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
252adantr 479 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝑅 Er 𝑋)
262, 13erref 7621 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2726adantr 479 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2813adantr 479 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑋)
29 elecg 7644 . . . . . . 7 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3028, 28, 29syl2anc 690 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
3127, 30mpbird 245 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
32 simpr 475 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3331, 32eleqtrd 2684 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3425, 32ereldm 7649 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴𝑋𝐵𝑋))
3528, 34mpbid 220 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑋)
36 elecg 7644 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3728, 35, 36syl2anc 690 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3833, 37mpbid 220 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3925, 38ersym 7613 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
4024, 39impbida 872 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  Vcvv 3167   class class class wbr 4572  Rel wrel 5028   Er wer 7598  [cec 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-br 4573  df-opab 4633  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-er 7601  df-ec 7603
This theorem is referenced by:  erth2  7651  erthi  7652  qliftfun  7691  eroveu  7701  eceqoveq  7712  enreceq  9738  prsrlem1  9744  ercpbllem  15972  orbsta  17510  sylow2blem3  17801  frgpnabllem2  18041  zndvds  19657  qustgpopn  21670  qustgphaus  21673  pi1xfrf  22587  pi1cof  22593  pstmxmet  29069  sconpi1  30276  topfneec2  31322
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