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Theorem frege98d 43743
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 43951. (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 5882 . . 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 1377 . 2 (𝜑𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8 trclfvcotrg 15052 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
98a1i 11 . . 3 (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
109ssbrd 5191 . 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 2106  Vcvv 3478  wss 3963   class class class wbr 5148  ccom 5693  cfv 6563  t+ctcl 15021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-trcl 15023
This theorem is referenced by: (None)
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