Theorem frege96d 44193

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 44403. (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 5810	. . 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 1386	. 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8		frege96d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
9		trclfvlb 14961	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10		coss1 5797	. . . . 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 14969	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
13	11, 12	sstrdi 3927	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
14	13	ssbrd 5115	. 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 2119  Vcvv 3431   ⊆ wss 3883   class class class wbr 5072   ∘ ccom 5622  ‘cfv 6485  t+ctcl 14938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-trcl 14940

This theorem is referenced by:  frege87d  44194  frege102d  44198  frege129d  44207