Theorem frege98d 44596

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 44804. (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 5848	. . 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 1405	. 2 ⊢ (𝜑 → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8		trclfvcotrg 15092	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
10	9	ssbrd 5148	. 2 ⊢ (𝜑 → (𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵 → 𝐴(t+‘𝑅)𝐵))
11	7, 10	mpd 16	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3899   class class class wbr 5103   ∘ ccom 5659  ‘cfv 6533  t+ctcl 15061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-fv 6541  df-trcl 15063

This theorem is used by: (None)