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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 39781 | . 2 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
4 | 1 | toptopon 21219 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | topgele 21232 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | |
6 | 4, 5 | sylbi 209 | . . 3 ⊢ (𝐽 ∈ Top → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
7 | 6 | simprd 488 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝒫 𝑋) |
8 | 3, 7 | fssd 6352 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ⊆ wss 3825 ∅c0 4173 𝒫 cpw 4416 {cpr 4437 ∪ cuni 4706 ⟶wf 6178 ‘cfv 6182 Topctop 21195 TopOnctopon 21212 intcnt 21319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-top 21196 df-topon 21213 df-ntr 21322 |
This theorem is referenced by: ntrelmap 39783 |
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