Theorem frege98d 38885

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 39094. (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 5527	. . 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 1501	. 2 ⊢ (𝜑 → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8		trclfvcotrg 14141	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
10	9	ssbrd 4918	. 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 2164  Vcvv 3414   ⊆ wss 3798   class class class wbr 4875   ∘ ccom 5350  ‘cfv 6127  t+ctcl 14110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fv 6135  df-trcl 14112

This theorem is referenced by: (None)