Theorem iscnrm 17392

Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypothesis

Ref	Expression
ist0.1	|- X = U. J

Assertion

Ref	Expression
iscnrm	|- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J ↾t x ) e. Nrm ) )

Distinct variable groups:   x, J   x, X

Proof of Theorem iscnrm

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4026	. . . . 5 |- ( j = J -> U. j = U. J )
2		ist0.1	. . . . 5 |- X = U. J
3	1, 2	syl6eqr 2488	. . . 4 |- ( j = J -> U. j = X )
4	3	pweqd 3806	. . 3 |- ( j = J -> ~P U. j = ~P X )
5		oveq1 6091	. . . 4 |- ( j = J -> ( j ↾t x ) = ( J ↾t x ) )
6	5	eleq1d 2504	. . 3 |- ( j = J -> ( ( j ↾t x ) e. Nrm <-> ( J ↾t x ) e. Nrm ) )
7	4, 6	raleqbidv 2918	. 2 |- ( j = J -> ( A. x e. ~P U. j ( j ↾t x ) e. Nrm <-> A. x e. ~P X ( J ↾t x ) e. Nrm ) )
8		df-cnrm 17387	. 2 |- CNrm = { j e. Top | A. x e. ~P U. j ( j ↾t x ) e. Nrm }
9	7, 8	elrab2 3096	1 |- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J ↾t x ) e. Nrm ) )

Syntax hints:   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1726   A.wral 2707   ~Pcpw 3801   U.cuni 4017  (class class class)co 6084   ↾_t crest 13653   Topctop 16963   Nrmcnrm 17379  CNrmccnrm 17380

This theorem is referenced by:  cnrmtop  17406  iscnrm2  17407  cnrmi  17429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-cnrm 17387