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Theorem iscnrm2 22712
Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
iscnrm2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝑋

Proof of Theorem iscnrm2
StepHypRef Expression
1 topontop 22285 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 eqid 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
32iscnrm 22697 . . . 4 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
43baib 537 . . 3 (𝐽 ∈ Top β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
51, 4syl 17 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
6 toponuni 22286 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76pweqd 4581 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
87raleqdv 3312 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm ↔ βˆ€π‘₯ ∈ 𝒫 βˆͺ 𝐽(𝐽 β†Ύt π‘₯) ∈ Nrm))
95, 8bitr4d 282 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐽 β†Ύt π‘₯) ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∈ wcel 2107  βˆ€wral 3061  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282  Nrmcnrm 22684  CNrmccnrm 22685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-topon 22283  df-cnrm 22692
This theorem is referenced by:  restcnrm  22736
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