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Theorem frege96d 40303
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 40514. (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 5726 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1375 . 2 (𝜑𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8 frege96d.r . . . . 5 (𝜑𝑅 ∈ V)
9 trclfvlb 14364 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10 coss1 5713 . . . . 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 14372 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
1311, 12sstrdi 3964 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
1413ssbrd 5095 . 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 2115  Vcvv 3480  wss 3919   class class class wbr 5052  ccom 5546  cfv 6343  t+ctcl 14341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-trcl 14343
This theorem is referenced by:  frege87d  40304  frege102d  40308  frege129d  40317
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