Theorem frege97d 40978

Description: If 𝐴 contains 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 97 of [Frege1879] p. 71. Compare with frege97 41186. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege97d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege97d.a	⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈))

Assertion

Ref	Expression
frege97d	⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Proof of Theorem frege97d

Step	Hyp	Ref	Expression
1		frege97d.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ V)
2		trclfvlb 14536	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		coss1 5709	. . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ ((t+‘𝑅) ∘ (t+‘𝑅)))
5		trclfvcotrg 14544	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
6	4, 5	sstrdi 3899	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7		imass1 5949	. . 3 ⊢ ((𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈))
9		frege97d.a	. . . 4 ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈))
10	9	imaeq2d 5914	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11		imaco 6095	. . 3 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
12	10, 11	eqtr4di 2789	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
13	8, 12, 9	3sstr4d 3934	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ⊆ wss 3853   “ cima 5539   ∘ ccom 5540  ‘cfv 6358  t+ctcl 14513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-int 4846  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fv 6366  df-trcl 14515

This theorem is referenced by: (None)