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Theorem frege96d 38998
Description: If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 39209. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege96d.r (𝜑𝑅 ∈ V)
frege96d.a (𝜑𝐴 ∈ V)
frege96d.b (𝜑𝐵 ∈ V)
frege96d.c (𝜑𝐶 ∈ V)
frege96d.ac (𝜑𝐴(t+‘𝑅)𝐶)
frege96d.cb (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
frege96d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege96d
StepHypRef Expression
1 frege96d.a . . 3 (𝜑𝐴 ∈ V)
2 frege96d.b . . 3 (𝜑𝐵 ∈ V)
3 frege96d.c . . 3 (𝜑𝐶 ∈ V)
4 frege96d.ac . . 3 (𝜑𝐴(t+‘𝑅)𝐶)
5 frege96d.cb . . 3 (𝜑𝐶𝑅𝐵)
6 brcogw 5536 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1446 . 2 (𝜑𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8 frege96d.r . . . . 5 (𝜑𝑅 ∈ V)
9 trclfvlb 14156 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10 coss1 5523 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
118, 9, 103syl 18 . . . 4 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
12 trclfvcotrg 14164 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
1311, 12syl6ss 3833 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
1413ssbrd 4929 . 2 (𝜑 → (𝐴(𝑅 ∘ (t+‘𝑅))𝐵𝐴(t+‘𝑅)𝐵))
157, 14mpd 15 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3398  wss 3792   class class class wbr 4886  ccom 5359  cfv 6135  t+ctcl 14133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-trcl 14135
This theorem is referenced by:  frege87d  38999  frege102d  39003  frege129d  39012
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