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Theorem vtoclr 5583
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 . . . . 5 Rel 𝑅
21brrelex12i 5575 . . . 4 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31brrelex2i 5577 . . . 4 (𝐵𝑅𝐶𝐶 ∈ V)
4 breq1 5036 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
54anbi1d 632 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦𝑦𝑅𝐶)))
6 breq1 5036 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑅𝐶𝐴𝑅𝐶))
75, 6imbi12d 348 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
87imbi2d 344 . . . . 5 (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
9 breq2 5037 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
10 breq1 5036 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
119, 10anbi12d 633 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1211imbi1d 345 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
1312imbi2d 344 . . . . 5 (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
14 breq2 5037 . . . . . . . 8 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
1514anbi2d 631 . . . . . . 7 (𝑧 = 𝐶 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐶)))
16 breq2 5037 . . . . . . 7 (𝑧 = 𝐶 → (𝑥𝑅𝑧𝑥𝑅𝐶))
1715, 16imbi12d 348 . . . . . 6 (𝑧 = 𝐶 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
18 vtoclr.2 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1917, 18vtoclg 3518 . . . . 5 (𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶))
208, 13, 19vtocl2g 3523 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
212, 3, 20syl2im 40 . . 3 (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2221imp 410 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
2322pm2.43i 52 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444   class class class wbr 5033  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by:  domtr  8549
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