Theorem frege91d 42435

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 42638. (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 14951	. . . 4 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
5	4	ssbrd 5190	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 → 𝐴(t+‘𝑅)𝐵))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3947   class class class wbr 5147  ‘cfv 6540  t+ctcl 14928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-trcl 14930

This theorem is referenced by:  frege102d  42438  frege129d  42447