Theorem frege98d 44330

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 44538. (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 5841	. . 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 1398	. 2 ⊢ (𝜑 → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8		trclfvcotrg 15030	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
10	9	ssbrd 5144	. 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 2143  Vcvv 3455   ⊆ wss 3905   class class class wbr 5101   ∘ ccom 5652  ‘cfv 6522  t+ctcl 14999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-iota 6478  df-fun 6524  df-fv 6530  df-trcl 15001

This theorem is referenced by: (None)