![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege109d | Structured version Visualization version GIF version |
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 39099. (Contributed by RP, 15-Jul-2020.) |
Ref | Expression |
---|---|
frege109d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
frege109d.a | ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) |
Ref | Expression |
---|---|
frege109d | ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege109d.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | trclfvlb 14126 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
3 | imass1 5741 | . . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈)) |
5 | coss1 5510 | . . . . . . 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 14134 | . . . . . 6 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
8 | 6, 7 | syl6ss 3839 | . . . . 5 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
9 | imass1 5741 | . . . . 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 4016 | . . 3 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈)) |
12 | ssun2 4004 | . . 3 ⊢ ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)) | |
13 | 11, 12 | syl6ss 3839 | . 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 5707 | . . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))) |
16 | imaundi 5786 | . . . 4 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) | |
17 | imaco 5881 | . . . . . 6 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈)) | |
18 | 17 | eqcomi 2834 | . . . . 5 ⊢ (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) |
19 | 18 | uneq2i 3991 | . . . 4 ⊢ ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) |
20 | 16, 19 | eqtri 2849 | . . 3 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) |
21 | 15, 20 | syl6eq 2877 | . 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))) |
22 | 13, 21, 14 | 3sstr4d 3873 | 1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 ⊆ wss 3798 “ cima 5345 ∘ ccom 5346 ‘cfv 6123 t+ctcl 14103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fv 6131 df-trcl 14105 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |