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Theorem frege109d 38883
 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 39099. (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 14126 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 imass1 5741 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5 coss1 5510 . . . . . . 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 14134 . . . . . 6 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
86, 7syl6ss 3839 . . . . 5 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9 imass1 5741 . . . . 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 4016 . . 3 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12 ssun2 4004 . . 3 ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
1311, 12syl6ss 3839 . 2 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14 frege109d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
1514imaeq2d 5707 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16 imaundi 5786 . . . 4 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17 imaco 5881 . . . . . 6 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1817eqcomi 2834 . . . . 5 (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
1918uneq2i 3991 . . . 4 ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2016, 19eqtri 2849 . . 3 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2115, 20syl6eq 2877 . 2 (𝜑 → (𝑅𝐴) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
2213, 21, 143sstr4d 3873 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798   “ cima 5345   ∘ ccom 5346  ‘cfv 6123  t+ctcl 14103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-trcl 14105 This theorem is referenced by: (None)
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