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Theorem eqvrelth 35291
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 35286 . . . . . . 7 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
32biimpa 469 . . . . . 6 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
41eqvreltr 35287 . . . . . . 7 (𝜑 → ((𝐵𝑅𝐴𝐴𝑅𝑥) → 𝐵𝑅𝑥))
54impl 448 . . . . . 6 (((𝜑𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
63, 5syldanl 592 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
71eqvreltr 35287 . . . . . 6 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝑥) → 𝐴𝑅𝑥))
87impl 448 . . . . 5 (((𝜑𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
96, 8impbida 788 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝐴𝑅𝑥𝐵𝑅𝑥))
10 vex 3412 . . . . 5 𝑥 ∈ V
11 eqvrelth.2 . . . . . 6 (𝜑𝐴 ∈ dom 𝑅)
1211adantr 473 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
13 elecg 8126 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ dom 𝑅) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
1410, 12, 13sylancr 578 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
15 eqvrelrel 35277 . . . . . . 7 ( EqvRel 𝑅 → Rel 𝑅)
161, 15syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
17 brrelex2 5450 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
1816, 17sylan 572 . . . . 5 ((𝜑𝐴𝑅𝐵) → 𝐵 ∈ V)
19 elecg 8126 . . . . 5 ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
2010, 18, 19sylancr 578 . . . 4 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅𝐵𝑅𝑥))
219, 14, 203bitr4d 303 . . 3 ((𝜑𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
2221eqrdv 2770 . 2 ((𝜑𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
231adantr 473 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → EqvRel 𝑅)
241, 11eqvrelref 35290 . . . . . . 7 (𝜑𝐴𝑅𝐴)
2524adantr 473 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
2611adantr 473 . . . . . . 7 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ dom 𝑅)
27 elecALTV 34971 . . . . . . 7 ((𝐴 ∈ dom 𝑅𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2826, 26, 27syl2anc 576 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅𝐴𝑅𝐴))
2925, 28mpbird 249 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30 simpr 477 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
3129, 30eleqtrd 2862 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
3230dmec2d 35007 . . . . . 6 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
3326, 32mpbid 224 . . . . 5 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ dom 𝑅)
34 elecALTV 34971 . . . . 5 ((𝐵 ∈ dom 𝑅𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3533, 26, 34syl2anc 576 . . . 4 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
3631, 35mpbid 224 . . 3 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
3723, 36eqvrelsym 35285 . 2 ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
3822, 37impbida 788 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  Vcvv 3409   class class class wbr 4923  dom cdm 5401  Rel wrel 5406  [cec 8081   EqvRel weqvrel 34914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ec 8085  df-refrel 35197  df-symrel 35225  df-trrel 35255  df-eqvrel 35265
This theorem is referenced by:  eqvrelthi  35293
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