Theorem frege98d 43744

Description: If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 43952. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege98d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege98d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege98d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege98d.ac	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)
frege98d.cb	⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵)

Assertion

Ref	Expression
frege98d	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege98d

Step	Hyp	Ref	Expression
1		frege98d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
2		frege98d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
3		frege98d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
4		frege98d.ac	. . 3 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)
5		frege98d.cb	. . 3 ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵)
6		brcogw 5853	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶 ∧ 𝐶(t+‘𝑅)𝐵)) → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
7	1, 2, 3, 4, 5, 6	syl32anc 1380	. 2 ⊢ (𝜑 → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8		trclfvcotrg 15040	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
10	9	ssbrd 5167	. 2 ⊢ (𝜑 → (𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵 → 𝐴(t+‘𝑅)𝐵))
11	7, 10	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   ∘ ccom 5663  ‘cfv 6536  t+ctcl 15009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fv 6544  df-trcl 15011

This theorem is referenced by: (None)