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Theorem frege97d 40451
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 40659. (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 14359 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 coss1 5690 . . . . 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 14367 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
64, 5sstrdi 3927 . . 3 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7 imass1 5931 . . 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 5896 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11 imaco 6071 . . 3 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1210, 11eqtr4di 2851 . 2 (𝜑 → (𝑅𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
138, 12, 93sstr4d 3962 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  Vcvv 3441  wss 3881  cima 5522  ccom 5523  cfv 6324  t+ctcl 14336
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-trcl 14338
This theorem is referenced by: (None)
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