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Theorem frege97d 39975
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 40184. (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 14356 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 coss1 5719 . . . . 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 14364 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
64, 5sstrdi 3976 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7 imass1 5957 . . 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 5922 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11 imaco 6097 . . 3 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1210, 11syl6eqr 2871 . 2 (𝜑 → (𝑅𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
138, 12, 93sstr4d 4011 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  cima 5551  ccom 5552  cfv 6348  t+ctcl 14333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-trcl 14335
This theorem is referenced by: (None)
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