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Theorem frege98d 43186
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 43394. (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 5873 . . 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 1375 . 2 (𝜑𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8 trclfvcotrg 15001 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
98a1i 11 . . 3 (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
109ssbrd 5193 . 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 2098  Vcvv 3471  wss 3947   class class class wbr 5150  ccom 5684  cfv 6551  t+ctcl 14970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fv 6559  df-trcl 14972
This theorem is referenced by: (None)
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