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Theorem frege109d 40239
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 40455. (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 14347 . . . . 5 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 imass1 5937 . . . . 5 (𝑅 ⊆ (t+‘𝑅) → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
41, 2, 33syl 18 . . . 4 (𝜑 → (𝑅𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5 coss1 5699 . . . . . . 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 14355 . . . . . 6 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
86, 7sstrdi 3955 . . . . 5 (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9 imass1 5937 . . . . 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 4138 . . 3 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12 ssun2 4125 . . 3 ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
1311, 12sstrdi 3955 . 2 (𝜑 → ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14 frege109d.a . . . 4 (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
1514imaeq2d 5902 . . 3 (𝜑 → (𝑅𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16 imaundi 5981 . . . 4 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17 imaco 6077 . . . . . 6 ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
1817eqcomi 2830 . . . . 5 (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
1918uneq2i 4112 . . . 4 ((𝑅𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2016, 19eqtri 2844 . . 3 (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
2115, 20syl6eq 2872 . 2 (𝜑 → (𝑅𝐴) = ((𝑅𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
2213, 21, 143sstr4d 3990 1 (𝜑 → (𝑅𝐴) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3471  cun 3908  wss 3910  cima 5531  ccom 5532  cfv 6328  t+ctcl 14324
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fv 6336  df-trcl 14326
This theorem is referenced by: (None)
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