Theorem cnrmi 21164

Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
cnrmi	|- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) e. Nrm )

Proof of Theorem cnrmi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	restin 20970	. 2 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) = ( J ↾t ( A i^i U. J ) ) )
3		inss2 3834	. . . . 5 |- ( A i^i U. J ) C_ U. J
4		inex1g 4801	. . . . . 6 |- ( A e. V -> ( A i^i U. J ) e. _V )
5		elpwg 4166	. . . . . 6 |- ( ( A i^i U. J ) e. _V -> ( ( A i^i U. J ) e. ~P U. J <-> ( A i^i U. J ) C_ U. J ) )
6	4, 5	syl 17	. . . . 5 |- ( A e. V -> ( ( A i^i U. J ) e. ~P U. J <-> ( A i^i U. J ) C_ U. J ) )
7	3, 6	mpbiri 248	. . . 4 |- ( A e. V -> ( A i^i U. J ) e. ~P U. J )
8	7	adantl 482	. . 3 |- ( ( J e. CNrm /\ A e. V ) -> ( A i^i U. J ) e. ~P U. J )
9	1	iscnrm 21127	. . . . 5 |- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P U. J ( J ↾t x ) e. Nrm ) )
10	9	simprbi 480	. . . 4 |- ( J e. CNrm -> A. x e. ~P U. J ( J ↾t x ) e. Nrm )
11	10	adantr 481	. . 3 |- ( ( J e. CNrm /\ A e. V ) -> A. x e. ~P U. J ( J ↾t x ) e. Nrm )
12		oveq2 6658	. . . . 5 |- ( x = ( A i^i U. J ) -> ( J ↾t x ) = ( J ↾t ( A i^i U. J ) ) )
13	12	eleq1d 2686	. . . 4 |- ( x = ( A i^i U. J ) -> ( ( J ↾t x ) e. Nrm <-> ( J ↾t ( A i^i U. J ) ) e. Nrm ) )
14	13	rspcv 3305	. . 3 |- ( ( A i^i U. J ) e. ~P U. J -> ( A. x e. ~P U. J ( J ↾t x ) e. Nrm -> ( J ↾t ( A i^i U. J ) ) e. Nrm ) )
15	8, 11, 14	sylc 65	. 2 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t ( A i^i U. J ) ) e. Nrm )
16	2, 15	eqeltrd 2701	1 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) e. Nrm )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083  df-cnrm 21122

This theorem is referenced by:  cnrmnrm  21165  restcnrm  21166