Theorem frege96d 40114

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 40325. (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 5739	. . 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 1374	. 2 ⊢ (𝜑 → 𝐴(𝑅 ∘ (t+‘𝑅))𝐵)
8		frege96d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
9		trclfvlb 14368	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
10		coss1 5726	. . . . 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 14376	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
13	11, 12	sstrdi 3979	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
14	13	ssbrd 5109	. 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 2114  Vcvv 3494   ⊆ wss 3936   class class class wbr 5066   ∘ ccom 5559  ‘cfv 6355  t+ctcl 14345

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fv 6363  df-trcl 14347

This theorem is referenced by:  frege87d  40115  frege102d  40119  frege129d  40128