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 = ∪ 𝐽
hauseqcn.k	⊢ (𝜑 → 𝐾 ∈ Haus)
hauseqcn.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
hauseqcn.g	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
hauseqcn.e	⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
hauseqcn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
hauseqcn.c	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)

Assertion

Ref	Expression
hauseqcn	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem hauseqcn

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

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  ∪ 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