Theorem cnindis 19666

Description: Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnindis	|- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J Cn { (/) , A } ) = ( A ^m X ) )

Proof of Theorem cnindis

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpri 4034	. . . . . . 7 |- ( x e. { (/) , A } -> ( x = (/) \/ x = A ) )
2		topontop 19300	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
3	2	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> J e. Top )
4		0opn 19286	. . . . . . . . . 10 |- ( J e. Top -> (/) e. J )
5	3, 4	syl 16	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> (/) e. J )
6		imaeq2 5323	. . . . . . . . . . 11 |- ( x = (/) -> ( `' f " x ) = ( `' f " (/) ) )
7		ima0 5342	. . . . . . . . . . 11 |- ( `' f " (/) ) = (/)
8	6, 7	syl6eq 2500	. . . . . . . . . 10 |- ( x = (/) -> ( `' f " x ) = (/) )
9	8	eleq1d 2512	. . . . . . . . 9 |- ( x = (/) -> ( ( `' f " x ) e. J <-> (/) e. J ) )
10	5, 9	syl5ibrcom 222	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = (/) -> ( `' f " x ) e. J ) )
11		fimacnv 6004	. . . . . . . . . . 11 |- ( f : X --> A -> ( `' f " A ) = X )
12	11	adantl 466	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) = X )
13		toponmax 19302	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X e. J )
14	13	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> X e. J )
15	12, 14	eqeltrd 2531	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) e. J )
16		imaeq2 5323	. . . . . . . . . 10 |- ( x = A -> ( `' f " x ) = ( `' f " A ) )
17	16	eleq1d 2512	. . . . . . . . 9 |- ( x = A -> ( ( `' f " x ) e. J <-> ( `' f " A ) e. J ) )
18	15, 17	syl5ibrcom 222	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = A -> ( `' f " x ) e. J ) )
19	10, 18	jaod 380	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( ( x = (/) \/ x = A ) -> ( `' f " x ) e. J ) )
20	1, 19	syl5 32	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x e. { (/) , A } -> ( `' f " x ) e. J ) )
21	20	ralrimiv 2855	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> A. x e. { (/) , A } ( `' f " x ) e. J )
22	21	ex 434	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f : X --> A -> A. x e. { (/) , A } ( `' f " x ) e. J ) )
23	22	pm4.71d 634	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f : X --> A <-> ( f : X --> A /\ A. x e. { (/) , A } ( `' f " x ) e. J ) ) )
24		id 22	. . . 4 |- ( A e. V -> A e. V )
25		elmapg 7435	. . . 4 |- ( ( A e. V /\ X e. J ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
26	24, 13, 25	syl2anr 478	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
27		indistopon 19375	. . . 4 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )
28		iscn 19609	. . . 4 |- ( ( J e. (TopOn ` X ) /\ { (/) , A } e. (TopOn ` A ) ) -> ( f e. ( J Cn { (/) , A } ) <-> ( f : X --> A /\ A. x e. { (/) , A } ( `' f " x ) e. J ) ) )
29	27, 28	sylan2 474	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( J Cn { (/) , A } ) <-> ( f : X --> A /\ A. x e. { (/) , A } ( `' f " x ) e. J ) ) )
30	23, 26, 29	3bitr4rd 286	. 2 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( J Cn { (/) , A } ) <-> f e. ( A ^m X ) ) )
31	30	eqrdv 2440	1 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J Cn { (/) , A } ) = ( A ^m X ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1383   e. wcel 1804   A.wral 2793   (/)c0 3770   {cpr 4016   `'ccnv 4988   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281   ^m cmap 7422   Topctop 19267  TopOnctopon 19268   Cn ccn 19598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-map 7424  df-top 19272  df-topon 19275  df-cn 19601

This theorem is referenced by:  indishmph  20172  indistgp  20472  indispcon  28552