Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege96d Structured version   Visualization version   GIF version

Theorem frege96d 36859
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 37072. (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 5196 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1325 . 2 (𝜑𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8 frege96d.r . . . . 5 (𝜑𝑅 ∈ V)
9 trclfvlb 13539 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10 coss1 5183 . . . . 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 13547 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
1311, 12syl6ss 3575 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
1413ssbrd 4616 . 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 1975  Vcvv 3168  wss 3535   class class class wbr 4573  ccom 5028  cfv 5786  t+ctcl 13514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-iota 5750  df-fun 5788  df-fv 5794  df-trcl 13516
This theorem is referenced by:  frege87d  36860  frege102d  36864  frege129d  36873
  Copyright terms: Public domain W3C validator