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Theorem ertr 8296
Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
Assertion
Ref Expression
ertr (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Proof of Theorem ertr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ersymb.1 . . . . . . 7 (𝜑𝑅 Er 𝑋)
2 errel 8290 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 17 . . . . . 6 (𝜑 → Rel 𝑅)
4 simpr 487 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5 brrelex1 5598 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
63, 4, 5syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐵 ∈ V)
7 simpr 487 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
8 breq2 5061 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
9 breq1 5060 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
108, 9anbi12d 632 . . . . 5 (𝑥 = 𝐵 → ((𝐴𝑅𝑥𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
116, 7, 10spcedv 3597 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶))
12 simpl 485 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐵)
13 brrelex1 5598 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
143, 12, 13syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴 ∈ V)
15 brrelex2 5599 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
163, 4, 15syl2an 597 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐶 ∈ V)
17 brcog 5730 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1814, 16, 17syl2anc 586 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1911, 18mpbird 259 . . 3 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴(𝑅𝑅)𝐶)
2019ex 415 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴(𝑅𝑅)𝐶))
21 df-er 8281 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
2221simp3bi 1141 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
231, 22syl 17 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
2423unssbd 4162 . . 3 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2524ssbrd 5100 . 2 (𝜑 → (𝐴(𝑅𝑅)𝐶𝐴𝑅𝐶))
2620, 25syld 47 1 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  Vcvv 3493  cun 3932  wss 3934   class class class wbr 5057  ccnv 5547  dom cdm 5548  ccom 5552  Rel wrel 5553   Er wer 8278
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-co 5557  df-er 8281
This theorem is referenced by:  ertrd  8297  erth  8330  iiner  8361  entr  8553  efginvrel2  18845  efgsrel  18852
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