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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsss | Structured version Visualization version GIF version |
Description: If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.) |
Ref | Expression |
---|---|
ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
ntrclsfv.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
ntrclsfv.t | ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclsss | ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . 4 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | ntrclsfv.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
5 | 1, 2, 3, 4 | ntrclsfv 37377 | . . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
6 | ntrclsfv.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) | |
7 | 1, 2, 3, 6 | ntrclsfv 37377 | . . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) |
8 | 5, 7 | sseq12d 3597 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))) |
9 | 1, 2, 3 | ntrclskex 37372 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
10 | 9 | ancli 572 | . . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))) |
11 | elmapi 7765 | . . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵) |
13 | 2, 3 | ntrclsrcomplex 37353 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵) |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵) |
15 | 12, 14 | ffvelrnd 6268 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵) |
16 | 15 | elpwid 4118 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵) |
17 | 2, 3 | ntrclsrcomplex 37353 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
18 | 17 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
19 | 12, 18 | ffvelrnd 6268 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵) |
20 | 19 | elpwid 4118 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) |
21 | 16, 20 | jca 553 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)) |
22 | sscon34b 37337 | . . 3 ⊢ (((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))) | |
23 | 10, 21, 22 | 3syl 18 | . 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))) |
24 | 8, 23 | bitr4d 270 | 1 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 |
This theorem is referenced by: (None) |
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