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Theorem ntrelmap 39262
Description: The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
Hypotheses
Ref Expression
ntrrn.x 𝑋 = 𝐽
ntrrn.i 𝐼 = (int‘𝐽)
Assertion
Ref Expression
ntrelmap (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))

Proof of Theorem ntrelmap
StepHypRef Expression
1 ntrrn.x . . 3 𝑋 = 𝐽
2 ntrrn.i . . 3 𝐼 = (int‘𝐽)
31, 2ntrf2 39261 . 2 (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋)
41topopn 21088 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
54pwexd 5081 . . 3 (𝐽 ∈ Top → 𝒫 𝑋 ∈ V)
65, 5elmapd 8141 . 2 (𝐽 ∈ Top → (𝐼 ∈ (𝒫 𝑋𝑚 𝒫 𝑋) ↔ 𝐼:𝒫 𝑋⟶𝒫 𝑋))
73, 6mpbird 249 1 (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋𝑚 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)
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