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Theorem ertr 6410
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 6404 . . . . . . 7 (𝑅 Er 𝑋 → Rel 𝑅)
31, 2syl 14 . . . . . 6 (𝜑 → Rel 𝑅)
4 simpr 109 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5 brrelex 4547 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
63, 4, 5syl2an 285 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐵 ∈ V)
7 simpr 109 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴𝑅𝐵𝐵𝑅𝐶))
8 breq2 3901 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
9 breq1 3900 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝑅𝐶𝐵𝑅𝐶))
108, 9anbi12d 462 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑅𝑥𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1110spcegv 2746 . . . . 5 (𝐵 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
126, 7, 11sylc 62 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶))
13 simpl 108 . . . . . 6 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐵)
14 brrelex 4547 . . . . . 6 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
153, 13, 14syl2an 285 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴 ∈ V)
16 brrelex2 4548 . . . . . 6 ((Rel 𝑅𝐵𝑅𝐶) → 𝐶 ∈ V)
173, 4, 16syl2an 285 . . . . 5 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐶 ∈ V)
18 brcog 4674 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
1915, 17, 18syl2anc 406 . . . 4 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → (𝐴(𝑅𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥𝑥𝑅𝐶)))
2012, 19mpbird 166 . . 3 ((𝜑 ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴(𝑅𝑅)𝐶)
2120ex 114 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴(𝑅𝑅)𝐶))
22 df-er 6395 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
2322simp3bi 981 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
241, 23syl 14 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
2524unssbd 3222 . . 3 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2625ssbrd 3939 . 2 (𝜑 → (𝐴(𝑅𝑅)𝐶𝐴𝑅𝐶))
2721, 26syld 45 1 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  cun 3037  wss 3039   class class class wbr 3897  ccnv 4506  dom cdm 4507  ccom 4511  Rel wrel 4512   Er wer 6392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-co 4516  df-er 6395
This theorem is referenced by:  ertrd  6411  erth  6439  iinerm  6467  entr  6644
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