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	⊢ (𝜑 → 𝑅 ∈ V)
frege109d.a	⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))

Assertion

Ref	Expression
frege109d	⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Proof of Theorem frege109d

Step	Hyp	Ref	Expression
1		frege109d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
2		trclfvlb 15047	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		imass1 6119	. . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5		coss1 5866	. . . . . . 7 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
6	1, 2, 5	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
7		trclfvcotrg 15055	. . . . . 6 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
8	6, 7	sstrdi 3996	. . . . 5 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9		imass1 6119	. . . . 5 ⊢ ((𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
11	4, 10	unssd 4192	. . 3 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12		ssun2 4179	. . 3 ⊢ ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
13	11, 12	sstrdi 3996	. 2 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
14		frege109d.a	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))
15	14	imaeq2d 6078	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16		imaundi 6169	. . . 4 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17		imaco 6271	. . . . . 6 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
18	17	eqcomi 2746	. . . . 5 ⊢ (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
19	18	uneq2i 4165	. . . 4 ⊢ ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
20	16, 19	eqtri 2765	. . 3 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
21	15, 20	eqtrdi 2793	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
22	13, 21, 14	3sstr4d 4039	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

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)