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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 41686 | . 2 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
4 | 1 | toptopon 22047 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | topgele 22060 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | |
6 | 4, 5 | sylbi 216 | . . 3 ⊢ (𝐽 ∈ Top → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
7 | 6 | simprd 495 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝒫 𝑋) |
8 | 3, 7 | fssd 6614 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 ∅c0 4261 𝒫 cpw 4538 {cpr 4568 ∪ cuni 4844 ⟶wf 6426 ‘cfv 6430 Topctop 22023 TopOnctopon 22040 intcnt 22149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-top 22024 df-topon 22041 df-ntr 22152 |
This theorem is referenced by: ntrelmap 41688 |
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