Theorem frege98d 43749

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 43957. (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 5835	. . 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 14989	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
9	8	a1i 11	. . 3 ⊢ (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
10	9	ssbrd 5153	. 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 3450   ⊆ wss 3917   class class class wbr 5110   ∘ ccom 5645  ‘cfv 6514  t+ctcl 14958

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-trcl 14960

This theorem is referenced by: (None)