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Theorem frege98d 37565
Description: If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 37776. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege98d.a (𝜑𝐴 ∈ V)
frege98d.b (𝜑𝐵 ∈ V)
frege98d.c (𝜑𝐶 ∈ V)
frege98d.ac (𝜑𝐴(t+‘𝑅)𝐶)
frege98d.cb (𝜑𝐶(t+‘𝑅)𝐵)
Assertion
Ref Expression
frege98d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege98d
StepHypRef Expression
1 frege98d.a . . 3 (𝜑𝐴 ∈ V)
2 frege98d.b . . 3 (𝜑𝐵 ∈ V)
3 frege98d.c . . 3 (𝜑𝐶 ∈ V)
4 frege98d.ac . . 3 (𝜑𝐴(t+‘𝑅)𝐶)
5 frege98d.cb . . 3 (𝜑𝐶(t+‘𝑅)𝐵)
6 brcogw 5260 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶(t+‘𝑅)𝐵)) → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1331 . 2 (𝜑𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8 trclfvcotrg 13707 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
98a1i 11 . . 3 (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
109ssbrd 4666 . 2 (𝜑 → (𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵𝐴(t+‘𝑅)𝐵))
117, 10mpd 15 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3190  wss 3560   class class class wbr 4623  ccom 5088  cfv 5857  t+ctcl 13674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-trcl 13676
This theorem is referenced by: (None)
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