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Theorem vtoclr 4631
 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
21brrelex1i 4626 . . . . 5 (𝐴𝑅𝐵𝐴 ∈ V)
31brrelex2i 4627 . . . . 5 (𝐴𝑅𝐵𝐵 ∈ V)
42, 3jca 304 . . . 4 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
51brrelex2i 4627 . . . 4 (𝐵𝑅𝐶𝐶 ∈ V)
6 breq1 3968 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
76anbi1d 461 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦𝑦𝑅𝐶)))
8 breq1 3968 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑅𝐶𝐴𝑅𝐶))
97, 8imbi12d 233 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
109imbi2d 229 . . . . 5 (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11 breq2 3969 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
12 breq1 3968 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
1311, 12anbi12d 465 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1413imbi1d 230 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
1514imbi2d 229 . . . . 5 (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16 breq2 3969 . . . . . . . 8 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
1716anbi2d 460 . . . . . . 7 (𝑧 = 𝐶 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐶)))
18 breq2 3969 . . . . . . 7 (𝑧 = 𝐶 → (𝑥𝑅𝑧𝑥𝑅𝐶))
1917, 18imbi12d 233 . . . . . 6 (𝑧 = 𝐶 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20 vtoclr.2 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
2119, 20vtoclg 2772 . . . . 5 (𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶))
2210, 15, 21vtocl2g 2776 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
234, 5, 22syl2im 38 . . 3 (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2423imp 123 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
2524pm2.43i 49 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  Vcvv 2712   class class class wbr 3965  Rel wrel 4588 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590 This theorem is referenced by:  domtr  6723
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