Theorem frege109d 44113

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 44328. (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 14943	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		imass1 6068	. . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
5		coss1 5812	. . . . . . 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 14951	. . . . . 6 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
8	6, 7	sstrdi 3948	. . . . 5 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9		imass1 6068	. . . . 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 4146	. . 3 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈))
12		ssun2 4133	. . 3 ⊢ ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))
13	11, 12	sstrdi 3948	. 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 6027	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))))
16		imaundi 6115	. . . 4 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈)))
17		imaco 6217	. . . . . 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 4119	. . . 4 ⊢ ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
20	16, 19	eqtri 2760	. . 3 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
21	15, 20	eqtrdi 2788	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)))
22	13, 21, 14	3sstr4d 3991	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903   “ cima 5635   ∘ ccom 5636  ‘cfv 6500  t+ctcl 14920

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-trcl 14922

This theorem is referenced by: (None)