Theorem iscnrm 17349

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 3992	. . . . 5 |- ( j = J -> U. j = U. J )
2		ist0.1	. . . . 5 |- X = U. J
3	1, 2	syl6eqr 2462	. . . 4 |- ( j = J -> U. j = X )
4	3	pweqd 3772	. . 3 |- ( j = J -> ~P U. j = ~P X )
5		oveq1 6055	. . . 4 |- ( j = J -> ( j ↾t x ) = ( J ↾t x ) )
6	5	eleq1d 2478	. . 3 |- ( j = J -> ( ( j ↾t x ) e. Nrm <-> ( J ↾t x ) e. Nrm ) )
7	4, 6	raleqbidv 2884	. 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 17344	. 2 |- CNrm = { j e. Top | A. x e. ~P U. j ( j ↾t x ) e. Nrm }
9	7, 8	elrab2 3062	1 |- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J ↾t x ) e. Nrm ) )

Syntax hints:   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2674   ~Pcpw 3767   U.cuni 3983  (class class class)co 6048   ↾_t crest 13611   Topctop 16921   Nrmcnrm 17336  CNrmccnrm 17337

This theorem is referenced by:  cnrmtop  17363  iscnrm2  17364  cnrmi  17386

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-cnrm 17344