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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrf2 | Structured version Visualization version GIF version |
Description: The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.) |
Ref | Expression |
---|---|
ntrrn.x | ⊢ 𝑋 = ∪ 𝐽 |
ntrrn.i | ⊢ 𝐼 = (int‘𝐽) |
Ref | Expression |
---|---|
ntrf2 | ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrrn.x | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | ntrrn.i | . . 3 ⊢ 𝐼 = (int‘𝐽) | |
3 | 1, 2 | ntrf 41351 | . 2 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
4 | 1 | toptopon 21768 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | topgele 21781 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | |
6 | 4, 5 | sylbi 220 | . . 3 ⊢ (𝐽 ∈ Top → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
7 | 6 | simprd 499 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝒫 𝑋) |
8 | 3, 7 | fssd 6541 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 {cpr 4529 ∪ cuni 4805 ⟶wf 6354 ‘cfv 6358 Topctop 21744 TopOnctopon 21761 intcnt 21868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-top 21745 df-topon 21762 df-ntr 21871 |
This theorem is referenced by: ntrelmap 41353 |
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