Theorem frege109d 40458

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 40673. (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 14359	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		imass1 5931	. . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5		coss1 5690	. . . . . . 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 14367	. . . . . 6 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
8	6, 7	sstrdi 3927	. . . . 5 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9		imass1 5931	. . . . 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 4113	. . 3 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12		ssun2 4100	. . 3 ⊢ ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
13	11, 12	sstrdi 3927	. 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 5896	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16		imaundi 5975	. . . 4 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17		imaco 6071	. . . . . 6 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
18	17	eqcomi 2807	. . . . 5 ⊢ (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)
19	18	uneq2i 4087	. . . 4 ⊢ ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
20	16, 19	eqtri 2821	. . 3 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
21	15, 20	eqtrdi 2849	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
22	13, 21, 14	3sstr4d 3962	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881   “ cima 5522   ∘ ccom 5523  ‘cfv 6324  t+ctcl 14336

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-trcl 14338

This theorem is referenced by: (None)