Theorem frege96d 43990

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 44200. (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 5817	. . 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 1380	. 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8		frege96d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
9		trclfvlb 14931	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10		coss1 5804	. . . . 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 14939	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
13	11, 12	sstrdi 3946	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
14	13	ssbrd 5141	. 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 2113  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098   ∘ ccom 5628  ‘cfv 6492  t+ctcl 14908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-trcl 14910

This theorem is referenced by:  frege87d  43991  frege102d  43995  frege129d  44004