Theorem frege97d 44329

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 44537. (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 15022	. . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3		coss1 5828	. . . . 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 15030	. . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
6	4, 5	sstrdi 3949	. . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
7		imass1 6091	. . 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 6050	. . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈)))
11		imaco 6239	. . 3 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈))
12	10, 11	eqtr4di 2816	. 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))
13	8, 12, 9	3sstr4d 3992	1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905   “ cima 5651   ∘ ccom 5652  ‘cfv 6522  t+ctcl 14999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fv 6530  df-trcl 15001

This theorem is referenced by: (None)