Theorem frege91d 40977

Description: If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 41180. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege91d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege91d.ac	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
frege91d	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege91d

Step	Hyp	Ref	Expression
1		frege91d.ac	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		frege91d.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
3		trclfvlb 14536	. . . 4 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
5	4	ssbrd 5082	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 → 𝐴(t+‘𝑅)𝐵))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3398   ⊆ wss 3853   class class class wbr 5039  ‘cfv 6358  t+ctcl 14513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-int 4846  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-iota 6316  df-fun 6360  df-fv 6366  df-trcl 14515

This theorem is referenced by:  frege102d  40980  frege129d  40989