Theorem frege91d 44204

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 44407. (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 14962	. . . 4 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
5	4	ssbrd 5116	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 → 𝐴(t+‘𝑅)𝐵))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883   class class class wbr 5073  ‘cfv 6486  t+ctcl 14939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-trcl 14941

This theorem is referenced by:  frege102d  44207  frege129d  44216