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Theorem frege109d 43110
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 43325. (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 14979 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 imass1 6099 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5 coss1 5852 . . . . . . 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 14987 . . . . . 6 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
86, 7sstrdi 3990 . . . . 5 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9 imass1 6099 . . . . 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 4182 . . 3 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12 ssun2 4169 . . 3 ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
1311, 12sstrdi 3990 . 2 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14 frege109d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
1514imaeq2d 6057 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16 imaundi 6148 . . . 4 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17 imaco 6249 . . . . . 6 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1817eqcomi 2736 . . . . 5 (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
1918uneq2i 4156 . . . 4 ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2016, 19eqtri 2755 . . 3 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2115, 20eqtrdi 2783 . 2 (𝜑 → (𝑅𝐴) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
2213, 21, 143sstr4d 4025 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3469  cun 3942  wss 3944  cima 5675  ccom 5676  cfv 6542  t+ctcl 14956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-trcl 14958
This theorem is referenced by: (None)
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