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Theorem vtoclr 4474
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 4468 . . . . 5 (𝐴𝑅𝐵𝐴 ∈ V)
31brrelex2i 4471 . . . . 5 (𝐴𝑅𝐵𝐵 ∈ V)
42, 3jca 300 . . . 4 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
51brrelex2i 4471 . . . 4 (𝐵𝑅𝐶𝐶 ∈ V)
6 breq1 3840 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
76anbi1d 453 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝑦𝑦𝑅𝐶)))
8 breq1 3840 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑅𝐶𝐴𝑅𝐶))
97, 8imbi12d 232 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶) ↔ ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)))
109imbi2d 228 . . . . 5 (𝑥 = 𝐴 → ((𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶))))
11 breq2 3841 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
12 breq1 3840 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
1311, 12anbi12d 457 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐶) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
1413imbi1d 229 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
1514imbi2d 228 . . . . 5 (𝑦 = 𝐵 → ((𝐶 ∈ V → ((𝐴𝑅𝑦𝑦𝑅𝐶) → 𝐴𝑅𝐶)) ↔ (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
16 breq2 3841 . . . . . . . 8 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
1716anbi2d 452 . . . . . . 7 (𝑧 = 𝐶 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐶)))
18 breq2 3841 . . . . . . 7 (𝑧 = 𝐶 → (𝑥𝑅𝑧𝑥𝑅𝐶))
1917, 18imbi12d 232 . . . . . 6 (𝑧 = 𝐶 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶)))
20 vtoclr.2 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
2119, 20vtoclg 2679 . . . . 5 (𝐶 ∈ V → ((𝑥𝑅𝑦𝑦𝑅𝐶) → 𝑥𝑅𝐶))
2210, 15, 21vtocl2g 2683 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ V → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
234, 5, 22syl2im 38 . . 3 (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2423imp 122 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
2524pm2.43i 48 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619   class class class wbr 3837  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by:  domtr  6482
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