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Theorem frege98d 37064
 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 37275. (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 5212 . . 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 1326 . 2 (𝜑𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8 trclfvcotrg 13605 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
98a1i 11 . . 3 (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
109ssbrd 4626 . 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 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ∘ ccom 5042  ‘cfv 5804  t+ctcl 13572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-trcl 13574 This theorem is referenced by: (None)
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