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Theorem frege97d 37564
Description: If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 37775. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege97d.r (𝜑𝑅 ∈ V)
frege97d.a (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))
Assertion
Ref Expression
frege97d (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Proof of Theorem frege97d
StepHypRef Expression
1 frege97d.r . . . . 5 (𝜑𝑅 ∈ V)
2 trclfvlb 13699 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 coss1 5247 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
5 trclfvcotrg 13707 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
64, 5syl6ss 3600 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7 imass1 5469 . . 3 ((𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
86, 7syl 17 . 2 (𝜑 → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
9 frege97d.a . . . 4 (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))
109imaeq2d 5435 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11 imaco 5609 . . 3 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1210, 11syl6eqr 2673 . 2 (𝜑 → (𝑅𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
138, 12, 93sstr4d 3633 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560  cima 5087  ccom 5088  cfv 5857  t+ctcl 13674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-trcl 13676
This theorem is referenced by: (None)
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