Theorem frege91d 44200

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 44403. (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 14965	. . . 4 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
5	4	ssbrd 5129	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 → 𝐴(t+‘𝑅)𝐵))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086  ‘cfv 6494  t+ctcl 14942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-iota 6450  df-fun 6496  df-fv 6502  df-trcl 14944

This theorem is referenced by:  frege102d  44203  frege129d  44212