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Theorem eqvrelth 37975
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.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypotheses
Ref Expression
eqvrelth.1 (𝜑 → EqvRel 𝑅)
eqvrelth.2 (𝜑𝐴 ∈ dom 𝑅)
Assertion
Ref Expression
eqvrelth (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem eqvrelth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqvrelth.1 . . . . . . . 8 (𝜑 → EqvRel 𝑅)
21eqvrelsymb 37970 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
32biimpa 476 . . . . . 6 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
41eqvreltr 37971 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
54impl 455 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
63, 5syldanl 601 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
71eqvreltr 37971 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
87impl 455 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
96, 8impbida 798 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
10 vex 3470 . . . . 5 𝑥 ∈ V
11 eqvrelth.2 . . . . . 6 (𝜑𝐴 ∈ dom 𝑅)
1211adantr 480 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
13 elecg 8743 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ dom 𝑅) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1410, 12, 13sylancr 586 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
15 eqvrelrel 37961 . . . . . . 7 ( EqvRel 𝑅 → Rel 𝑅)
161, 15syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
17 brrelex2 5721 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
1816, 17sylan 579 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
19 elecg 8743 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2010, 18, 19sylancr 586 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
219, 14, 203bitr4d 311 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2221eqrdv 2722 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
231adantr 480 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → EqvRel 𝑅)
241, 11eqvrelref 37974 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2524adantr 480 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2611adantr 480 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ dom 𝑅)
27 elecALTV 37628 . . . . . . 7 ((𝐴 ∈ dom 𝑅𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2826, 26, 27syl2anc 583 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2925, 28mpbird 257 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30 simpr 484 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3129, 30eleqtrd 2827 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3230dmec2d 37668 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
3326, 32mpbid 231 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ dom 𝑅)
34 elecALTV 37628 . . . . 5 ((𝐵 ∈ dom 𝑅𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3533, 26, 34syl2anc 583 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3631, 35mpbid 231 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3723, 36eqvrelsym 37969 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
3822, 37impbida 798 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3466   class class class wbr 5139  dom cdm 5667  Rel wrel 5672  [cec 8698   EqvRel weqvrel 37554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-refrel 37876  df-symrel 37908  df-trrel 37938  df-eqvrel 37949
This theorem is referenced by:  eqvrelthi  37977
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