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Theorem frege96d 43173
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 43383. (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 5865 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1376 . 2 (𝜑𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8 frege96d.r . . . . 5 (𝜑𝑅 ∈ V)
9 trclfvlb 14981 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10 coss1 5852 . . . . 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 14989 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
1311, 12sstrdi 3990 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
1413ssbrd 5185 . 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 2099  Vcvv 3470  wss 3945   class class class wbr 5142  ccom 5676  cfv 6542  t+ctcl 14958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-trcl 14960
This theorem is referenced by:  frege87d  43174  frege102d  43178  frege129d  43187
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