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Theorem vtoclr 4415
Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
vtoclr.1 Rel 𝑅
vtoclr.2 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
Assertion
Ref Expression
vtoclr ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑥,𝑧)

Proof of Theorem vtoclr
StepHypRef Expression
1 vtoclr.1 . . . . . 6 Rel 𝑅
21brrelexi 4411 . . . . 5 (𝐴𝑅𝐵𝐴 ∈ V)
31brrelex2i 4412 . . . . 5 (𝐴𝑅𝐵𝐵 ∈ V)
42, 3jca 294 . . . 4 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
51brrelex2i 4412 . . . 4 (𝐵𝑅𝐶𝐶 ∈ V)
6 breq1 3794 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
76anbi1d 446 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦𝑦𝑅𝐶)))
8 breq1 3794 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑅𝐶𝐴𝑅𝐶))
97, 8imbi12d 227 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
109imbi2d 223 . . . . 5 (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11 breq2 3795 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
12 breq1 3794 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
1311, 12anbi12d 450 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1413imbi1d 224 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
1514imbi2d 223 . . . . 5 (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16 breq2 3795 . . . . . . . 8 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
1716anbi2d 445 . . . . . . 7 (𝑧 = 𝐶 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐶)))
18 breq2 3795 . . . . . . 7 (𝑧 = 𝐶 → (𝑥𝑅𝑧𝑥𝑅𝐶))
1917, 18imbi12d 227 . . . . . 6 (𝑧 = 𝐶 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20 vtoclr.2 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
2119, 20vtoclg 2630 . . . . 5 (𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶))
2210, 15, 21vtocl2g 2634 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
234, 5, 22syl2im 38 . . 3 (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2423imp 119 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
2524pm2.43i 47 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574   class class class wbr 3791  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379
This theorem is referenced by:  domtr  6295
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