Theorem frege96d 41357

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 41567. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege96d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege96d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege96d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege96d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege96d.ac	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)
frege96d.cb	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
frege96d	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege96d

Step	Hyp	Ref	Expression
1		frege96d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
2		frege96d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
3		frege96d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
4		frege96d.ac	. . 3 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)
5		frege96d.cb	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
6		brcogw 5777	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶 ∧ 𝐶𝑅𝐵)) → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
7	1, 2, 3, 4, 5, 6	syl32anc 1377	. 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8		frege96d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
9		trclfvlb 14719	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10		coss1 5764	. . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
11	8, 9, 10	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
12		trclfvcotrg 14727	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
13	11, 12	sstrdi 3933	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
14	13	ssbrd 5117	. 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ (t+‘𝑅))𝐵 → 𝐴(t+‘𝑅)𝐵))
15	7, 14	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   ∘ ccom 5593  ‘cfv 6433  t+ctcl 14696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-trcl 14698

This theorem is referenced by:  frege87d  41358  frege102d  41362  frege129d  41371