![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrelmap | 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 |
---|---|
ntrelmap | ⊢ (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrrn.x | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | ntrrn.i | . . 3 ⊢ 𝐼 = (int‘𝐽) | |
3 | 1, 2 | ntrf2 39261 | . 2 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋) |
4 | 1 | topopn 21088 | . . . 4 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | 4 | pwexd 5081 | . . 3 ⊢ (𝐽 ∈ Top → 𝒫 𝑋 ∈ V) |
6 | 5, 5 | elmapd 8141 | . 2 ⊢ (𝐽 ∈ Top → (𝐼 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋) ↔ 𝐼:𝒫 𝑋⟶𝒫 𝑋)) |
7 | 3, 6 | mpbird 249 | 1 ⊢ (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 𝒫 cpw 4380 ∪ cuni 4660 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Topctop 21075 intcnt 21199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-top 21076 df-topon 21093 df-ntr 21202 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |