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Theorem frege96d 37867
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 38079. (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 5288 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1333 . 2 (𝜑𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8 frege96d.r . . . . 5 (𝜑𝑅 ∈ V)
9 trclfvlb 13743 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10 coss1 5275 . . . . 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 13751 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
1311, 12syl6ss 3613 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
1413ssbrd 4694 . 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 1989  Vcvv 3198  wss 3572   class class class wbr 4651  ccom 5116  cfv 5886  t+ctcl 13718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-iota 5849  df-fun 5888  df-fv 5894  df-trcl 13720
This theorem is referenced by:  frege87d  37868  frege102d  37872  frege129d  37881
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