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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 43325. (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 14979 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
3 | imass1 6099 | . . . . 5 ⊢ (𝑅 ⊆ (t+‘𝑅) → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ ((t+‘𝑅) “ 𝑈)) |
5 | coss1 5852 | . . . . . . 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 14987 | . . . . . 6 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | |
8 | 6, 7 | sstrdi 3990 | . . . . 5 ⊢ (𝜑 → (𝑅 ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
9 | imass1 6099 | . . . . 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 4182 | . . 3 ⊢ (𝜑 → ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) ⊆ ((t+‘𝑅) “ 𝑈)) |
12 | ssun2 4169 | . . 3 ⊢ ((t+‘𝑅) “ 𝑈) ⊆ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)) | |
13 | 11, 12 | sstrdi 3990 | . 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 6057 | . . 3 ⊢ (𝜑 → (𝑅 “ 𝐴) = (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))) |
16 | imaundi 6148 | . . . 4 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) | |
17 | imaco 6249 | . . . . . 6 ⊢ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) = (𝑅 “ ((t+‘𝑅) “ 𝑈)) | |
18 | 17 | eqcomi 2736 | . . . . 5 ⊢ (𝑅 “ ((t+‘𝑅) “ 𝑈)) = ((𝑅 ∘ (t+‘𝑅)) “ 𝑈) |
19 | 18 | uneq2i 4156 | . . . 4 ⊢ ((𝑅 “ 𝑈) ∪ (𝑅 “ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) |
20 | 16, 19 | eqtri 2755 | . . 3 ⊢ (𝑅 “ (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈)) |
21 | 15, 20 | eqtrdi 2783 | . 2 ⊢ (𝜑 → (𝑅 “ 𝐴) = ((𝑅 “ 𝑈) ∪ ((𝑅 ∘ (t+‘𝑅)) “ 𝑈))) |
22 | 13, 21, 14 | 3sstr4d 4025 | 1 ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ⊆ wss 3944 “ cima 5675 ∘ ccom 5676 ‘cfv 6542 t+ctcl 14956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-trcl 14958 |
This theorem is referenced by: (None) |
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