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Theorem cotr 4629
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cotr ((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))
Distinct variable group:   x,y,z,𝑅

Proof of Theorem cotr
StepHypRef Expression
1 df-co 4277 . . . 4 (𝑅𝑅) = {⟨x, z⟩ ∣ y(x𝑅y y𝑅z)}
21relopabi 4386 . . 3 Rel (𝑅𝑅)
3 ssrel 4351 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅)))
42, 3ax-mp 7 . 2 ((𝑅𝑅) ⊆ 𝑅xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅))
5 vex 2534 . . . . . . . 8 x V
6 vex 2534 . . . . . . . 8 z V
75, 6opelco 4430 . . . . . . 7 (⟨x, z (𝑅𝑅) ↔ y(x𝑅y y𝑅z))
8 df-br 3735 . . . . . . . 8 (x𝑅z ↔ ⟨x, z 𝑅)
98bicomi 123 . . . . . . 7 (⟨x, z 𝑅x𝑅z)
107, 9imbi12i 228 . . . . . 6 ((⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ (y(x𝑅y y𝑅z) → x𝑅z))
11 19.23v 1741 . . . . . 6 (y((x𝑅y y𝑅z) → x𝑅z) ↔ (y(x𝑅y y𝑅z) → x𝑅z))
1210, 11bitr4i 176 . . . . 5 ((⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ y((x𝑅y y𝑅z) → x𝑅z))
1312albii 1335 . . . 4 (z(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ zy((x𝑅y y𝑅z) → x𝑅z))
14 alcom 1343 . . . 4 (zy((x𝑅y y𝑅z) → x𝑅z) ↔ yz((x𝑅y y𝑅z) → x𝑅z))
1513, 14bitri 173 . . 3 (z(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ yz((x𝑅y y𝑅z) → x𝑅z))
1615albii 1335 . 2 (xz(⟨x, z (𝑅𝑅) → ⟨x, z 𝑅) ↔ xyz((x𝑅y y𝑅z) → x𝑅z))
174, 16bitri 173 1 ((𝑅𝑅) ⊆ 𝑅xyz((x𝑅y y𝑅z) → x𝑅z))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224  wex 1358   wcel 1370  wss 2890  cop 3349   class class class wbr 3734  ccom 4272  Rel wrel 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-co 4277
This theorem is referenced by:  xpidtr  4638  trin2  4639  dfer2  6014
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