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Theorem vtoclr 5333
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 5327 . . . . 5 (𝐴𝑅𝐵𝐴 ∈ V)
31brrelex2i 5328 . . . . 5 (𝐴𝑅𝐵𝐵 ∈ V)
42, 3jca 507 . . . 4 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
51brrelex2i 5328 . . . 4 (𝐵𝑅𝐶𝐶 ∈ V)
6 breq1 4811 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
76anbi1d 623 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦𝑦𝑅𝐶)))
8 breq1 4811 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑅𝐶𝐴𝑅𝐶))
97, 8imbi12d 335 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
109imbi2d 331 . . . . 5 (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11 breq2 4812 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
12 breq1 4811 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
1311, 12anbi12d 624 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1413imbi1d 332 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
1514imbi2d 331 . . . . 5 (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16 breq2 4812 . . . . . . . 8 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
1716anbi2d 622 . . . . . . 7 (𝑧 = 𝐶 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐶)))
18 breq2 4812 . . . . . . 7 (𝑧 = 𝐶 → (𝑥𝑅𝑧𝑥𝑅𝐶))
1917, 18imbi12d 335 . . . . . 6 (𝑧 = 𝐶 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20 vtoclr.2 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
2119, 20vtoclg 3417 . . . . 5 (𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶))
2210, 15, 21vtocl2g 3421 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
234, 5, 22syl2im 40 . . 3 (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2423imp 395 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
2524pm2.43i 52 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3349   class class class wbr 4808  Rel wrel 5281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-br 4809  df-opab 4871  df-xp 5282  df-rel 5283
This theorem is referenced by:  domtr  8212
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