Theorem hauseqcn 29941

Description: In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)

Hypotheses

Ref	Expression
hauseqcn.x	|- X = U. J
hauseqcn.k	|- ( ph -> K e. Haus )
hauseqcn.f	|- ( ph -> F e. ( J Cn K ) )
hauseqcn.g	|- ( ph -> G e. ( J Cn K ) )
hauseqcn.e	|- ( ph -> ( F |` A ) = ( G |` A ) )
hauseqcn.a	|- ( ph -> A C_ X )
hauseqcn.c	|- ( ph -> ( ( cls ` J ) ` A ) = X )

Assertion

Ref	Expression
hauseqcn	|- ( ph -> F = G )

Proof of Theorem hauseqcn

Step	Hyp	Ref	Expression
1		hauseqcn.x	. . 3 |- X = U. J
2		hauseqcn.f	. . . . . 6 |- ( ph -> F e. ( J Cn K ) )
3		cntop1 21044	. . . . . 6 |- ( F e. ( J Cn K ) -> J e. Top )
4	2, 3	syl 17	. . . . 5 |- ( ph -> J e. Top )
5		dmin 5332	. . . . . 6 |- dom ( F i^i G ) C_ ( dom F i^i dom G )
6		eqid 2622	. . . . . . . . . 10 |- U. J = U. J
7		eqid 2622	. . . . . . . . . 10 |- U. K = U. K
8	6, 7	cnf 21050	. . . . . . . . 9 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
9		fdm 6051	. . . . . . . . 9 |- ( F : U. J --> U. K -> dom F = U. J )
10	2, 8, 9	3syl 18	. . . . . . . 8 |- ( ph -> dom F = U. J )
11		hauseqcn.g	. . . . . . . . 9 |- ( ph -> G e. ( J Cn K ) )
12	6, 7	cnf 21050	. . . . . . . . 9 |- ( G e. ( J Cn K ) -> G : U. J --> U. K )
13		fdm 6051	. . . . . . . . 9 |- ( G : U. J --> U. K -> dom G = U. J )
14	11, 12, 13	3syl 18	. . . . . . . 8 |- ( ph -> dom G = U. J )
15	10, 14	ineq12d 3815	. . . . . . 7 |- ( ph -> ( dom F i^i dom G ) = ( U. J i^i U. J ) )
16		inidm 3822	. . . . . . 7 |- ( U. J i^i U. J ) = U. J
17	15, 16	syl6eq 2672	. . . . . 6 |- ( ph -> ( dom F i^i dom G ) = U. J )
18	5, 17	syl5sseq 3653	. . . . 5 |- ( ph -> dom ( F i^i G ) C_ U. J )
19		hauseqcn.e	. . . . . 6 |- ( ph -> ( F |` A ) = ( G |` A ) )
20		ffn 6045	. . . . . . . 8 |- ( F : U. J --> U. K -> F Fn U. J )
21	2, 8, 20	3syl 18	. . . . . . 7 |- ( ph -> F Fn U. J )
22		ffn 6045	. . . . . . . 8 |- ( G : U. J --> U. K -> G Fn U. J )
23	11, 12, 22	3syl 18	. . . . . . 7 |- ( ph -> G Fn U. J )
24		hauseqcn.a	. . . . . . . 8 |- ( ph -> A C_ X )
25	24, 1	syl6sseq 3651	. . . . . . 7 |- ( ph -> A C_ U. J )
26		fnreseql 6327	. . . . . . 7 |- ( ( F Fn U. J /\ G Fn U. J /\ A C_ U. J ) -> ( ( F |` A ) = ( G |` A ) <-> A C_ dom ( F i^i G ) ) )
27	21, 23, 25, 26	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( F |` A ) = ( G |` A ) <-> A C_ dom ( F i^i G ) ) )
28	19, 27	mpbid 222	. . . . 5 |- ( ph -> A C_ dom ( F i^i G ) )
29	6	clsss 20858	. . . . 5 |- ( ( J e. Top /\ dom ( F i^i G ) C_ U. J /\ A C_ dom ( F i^i G ) ) -> ( ( cls ` J ) ` A ) C_ ( ( cls ` J ) ` dom ( F i^i G ) ) )
30	4, 18, 28, 29	syl3anc 1326	. . . 4 |- ( ph -> ( ( cls ` J ) ` A ) C_ ( ( cls ` J ) ` dom ( F i^i G ) ) )
31		hauseqcn.c	. . . 4 |- ( ph -> ( ( cls ` J ) ` A ) = X )
32		hauseqcn.k	. . . . . 6 |- ( ph -> K e. Haus )
33	32, 2, 11	hauseqlcld 21449	. . . . 5 |- ( ph -> dom ( F i^i G ) e. ( Clsd ` J ) )
34		cldcls 20846	. . . . 5 |- ( dom ( F i^i G ) e. ( Clsd ` J ) -> ( ( cls ` J ) ` dom ( F i^i G ) ) = dom ( F i^i G ) )
35	33, 34	syl 17	. . . 4 |- ( ph -> ( ( cls ` J ) ` dom ( F i^i G ) ) = dom ( F i^i G ) )
36	30, 31, 35	3sstr3d 3647	. . 3 |- ( ph -> X C_ dom ( F i^i G ) )
37	1, 36	syl5eqssr 3650	. 2 |- ( ph -> U. J C_ dom ( F i^i G ) )
38		fneqeql2 6326	. . 3 |- ( ( F Fn U. J /\ G Fn U. J ) -> ( F = G <-> U. J C_ dom ( F i^i G ) ) )
39	21, 23, 38	syl2anc 693	. 2 |- ( ph -> ( F = G <-> U. J C_ dom ( F i^i G ) ) )
40	37, 39	mpbird 247	1 |- ( ph -> F = G )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698   Clsdccld 20820   clsccl 20822   Cn ccn 21028   Hauscha 21112

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-int 4476  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-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cls 20825  df-cn 21031  df-haus 21119  df-tx 21365

This theorem is referenced by:  rrhre  30065