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Theorem frege109d 43746
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 43961. (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 15043 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 imass1 6121 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5 coss1 5868 . . . . . . 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 15051 . . . . . 6 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
86, 7sstrdi 4007 . . . . 5 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9 imass1 6121 . . . . 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 4201 . . 3 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12 ssun2 4188 . . 3 ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
1311, 12sstrdi 4007 . 2 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14 frege109d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
1514imaeq2d 6079 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16 imaundi 6171 . . . 4 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17 imaco 6272 . . . . . 6 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1817eqcomi 2743 . . . . 5 (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
1918uneq2i 4174 . . . 4 ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2016, 19eqtri 2762 . . 3 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2115, 20eqtrdi 2790 . 2 (𝜑 → (𝑅𝐴) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
2213, 21, 143sstr4d 4042 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  wss 3962  cima 5691  ccom 5692  cfv 6562  t+ctcl 15020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-trcl 15022
This theorem is referenced by: (None)
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