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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege97d | Structured version Visualization version GIF version |
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 40304. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege97d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege97d.a | ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈)) |
Ref | Expression |
---|---|
frege97d | ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege97d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | trclfvlb 14367 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
3 | coss1 5725 | . . . . 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 14375 | . . . 4 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
6 | 4, 5 | sstrdi 3978 | . . 3 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
7 | imass1 5963 | . . 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 5928 | . . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ ((t+‘𝑅) “ 𝑈))) |
11 | imaco 6103 | . . 3 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈)) | |
12 | 10, 11 | syl6eqr 2874 | . 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) |
13 | 8, 12, 9 | 3sstr4d 4013 | 1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 “ cima 5557 ∘ ccom 5558 ‘cfv 6354 t+ctcl 14344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-trcl 14346 |
This theorem is referenced by: (None) |
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