Theorem iscnrm 21127

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 4444	. . . . 5 |- ( j = J -> U. j = U. J )
2		ist0.1	. . . . 5 |- X = U. J
3	1, 2	syl6eqr 2674	. . . 4 |- ( j = J -> U. j = X )
4	3	pweqd 4163	. . 3 |- ( j = J -> ~P U. j = ~P X )
5		oveq1 6657	. . . 4 |- ( j = J -> ( j ↾t x ) = ( J ↾t x ) )
6	5	eleq1d 2686	. . 3 |- ( j = J -> ( ( j ↾t x ) e. Nrm <-> ( J ↾t x ) e. Nrm ) )
7	4, 6	raleqbidv 3152	. 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 21122	. 2 |- CNrm = { j e. Top | A. x e. ~P U. j ( j ↾t x ) e. Nrm }
9	7, 8	elrab2 3366	1 |- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J ↾t x ) e. Nrm ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ~Pcpw 4158   U.cuni 4436  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   Nrmcnrm 21114  CNrmccnrm 21115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cnrm 21122

This theorem is referenced by:  cnrmtop  21141  iscnrm2  21142  cnrmi  21164