Theorem frege91d 43741

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 43944. (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 15044	. . . 4 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
5	4	ssbrd 5191	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 → 𝐴(t+‘𝑅)𝐵))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148  ‘cfv 6563  t+ctcl 15021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-trcl 15023

This theorem is referenced by:  frege102d  43744  frege129d  43753