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Theorem frege109d 43770
Description: If 𝐴 contains all elements of 𝑈 and 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 109 of [Frege1879] p. 74. Compare with frege109 43985. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege109d.r (𝜑𝑅 ∈ V)
frege109d.a (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
Assertion
Ref Expression
frege109d (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Proof of Theorem frege109d
StepHypRef Expression
1 frege109d.r . . . . 5 (𝜑𝑅 ∈ V)
2 trclfvlb 15047 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 imass1 6119 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5 coss1 5866 . . . . . . 7 (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
61, 2, 53syl 18 . . . . . 6 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
7 trclfvcotrg 15055 . . . . . 6 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
86, 7sstrdi 3996 . . . . 5 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9 imass1 6119 . . . . 5 ((𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
108, 9syl 17 . . . 4 (𝜑 → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
114, 10unssd 4192 . . 3 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12 ssun2 4179 . . 3 ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
1311, 12sstrdi 3996 . 2 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14 frege109d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
1514imaeq2d 6078 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16 imaundi 6169 . . . 4 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17 imaco 6271 . . . . . 6 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1817eqcomi 2746 . . . . 5 (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
1918uneq2i 4165 . . . 4 ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2016, 19eqtri 2765 . . 3 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2115, 20eqtrdi 2793 . 2 (𝜑 → (𝑅𝐴) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
2213, 21, 143sstr4d 4039 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  cima 5688  ccom 5689  cfv 6561  t+ctcl 15024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-trcl 15026
This theorem is referenced by: (None)
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