frege97d - Mathbox for Richard Penner

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Theorem frege97d 42488

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 42696. (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 14951	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		coss1 5853	. . . . 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 14959	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
6	4, 5	sstrdi 3993	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7		imass1 6097	. . 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 6057	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11		imaco 6247	. . 3 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
12	10, 11	eqtr4di 2790	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
13	8, 12, 9	3sstr4d 4028	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 “ cima 5678 ∘ ccom 5679 ‘cfv 6540 t+ctcl 14928

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fv 6548 df-trcl 14930

This theorem is referenced by: (None)

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