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Theorem eqvreltr 36462
Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
Hypothesis
Ref Expression
eqvreltr.1 (𝜑 → EqvRel 𝑅)
Assertion
Ref Expression
eqvreltr (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Proof of Theorem eqvreltr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqvreltr.1 . . . . . . 7 (𝜑 → EqvRel 𝑅)
2 eqvrelrel 36452 . . . . . . 7 ( EqvRel 𝑅 → Rel 𝑅)
31, 2syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
4 simpr 488 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5 brrelex1 5607 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
63, 4, 5syl2an 599 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐵 ∈ V)
7 simpr 488 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
8 breq2 5062 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
9 breq1 5061 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
108, 9anbi12d 634 . . . . 5 (𝑥 = 𝐵 → ((𝐴𝑅𝑥𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
116, 7, 10spcedv 3518 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶))
12 simpl 486 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐵)
13 brrelex1 5607 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
143, 12, 13syl2an 599 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴 ∈ V)
15 brrelex2 5608 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
163, 4, 15syl2an 599 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐶 ∈ V)
17 brcog 5740 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1814, 16, 17syl2anc 587 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1911, 18mpbird 260 . . 3 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴(𝑅𝑅)𝐶)
2019ex 416 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴(𝑅𝑅)𝐶))
21 dfeqvrel2 36445 . . . . . 6 ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
2221simplbi 501 . . . . 5 ( EqvRel 𝑅 → (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅))
2322simp3d 1146 . . . 4 ( EqvRel 𝑅 → (𝑅𝑅) ⊆ 𝑅)
241, 23syl 17 . . 3 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2524ssbrd 5101 . 2 (𝜑 → (𝐴(𝑅𝑅)𝐶𝐴𝑅𝐶))
2620, 25syld 47 1 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  Vcvv 3413  wss 3871   class class class wbr 5058   I cid 5459  ccnv 5555  dom cdm 5556  cres 5558  ccom 5560  Rel wrel 5561   EqvRel weqvrel 36092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-br 5059  df-opab 5121  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-refrel 36372  df-symrel 36400  df-trrel 36430  df-eqvrel 36440
This theorem is referenced by:  eqvreltrd  36463  eqvrelth  36466
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