Theorem cnindis 21096

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 4197	. . . . . . 7 |- ( x e. { (/) , A } -> ( x = (/) \/ x = A ) )
2		topontop 20718	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
3	2	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> J e. Top )
4		0opn 20709	. . . . . . . . . 10 |- ( J e. Top -> (/) e. J )
5	3, 4	syl 17	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> (/) e. J )
6		imaeq2 5462	. . . . . . . . . . 11 |- ( x = (/) -> ( `' f " x ) = ( `' f " (/) ) )
7		ima0 5481	. . . . . . . . . . 11 |- ( `' f " (/) ) = (/)
8	6, 7	syl6eq 2672	. . . . . . . . . 10 |- ( x = (/) -> ( `' f " x ) = (/) )
9	8	eleq1d 2686	. . . . . . . . 9 |- ( x = (/) -> ( ( `' f " x ) e. J <-> (/) e. J ) )
10	5, 9	syl5ibrcom 237	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = (/) -> ( `' f " x ) e. J ) )
11		fimacnv 6347	. . . . . . . . . . 11 |- ( f : X --> A -> ( `' f " A ) = X )
12	11	adantl 482	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) = X )
13		toponmax 20730	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X e. J )
14	13	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> X e. J )
15	12, 14	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) e. J )
16		imaeq2 5462	. . . . . . . . . 10 |- ( x = A -> ( `' f " x ) = ( `' f " A ) )
17	16	eleq1d 2686	. . . . . . . . 9 |- ( x = A -> ( ( `' f " x ) e. J <-> ( `' f " A ) e. J ) )
18	15, 17	syl5ibrcom 237	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = A -> ( `' f " x ) e. J ) )
19	10, 18	jaod 395	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( ( x = (/) \/ x = A ) -> ( `' f " x ) e. J ) )
20	1, 19	syl5 34	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x e. { (/) , A } -> ( `' f " x ) e. J ) )
21	20	ralrimiv 2965	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> A. x e. { (/) , A } ( `' f " x ) e. J )
22	21	ex 450	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f : X --> A -> A. x e. { (/) , A } ( `' f " x ) e. J ) )
23	22	pm4.71d 666	. . 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 7870	. . . 4 |- ( ( A e. V /\ X e. J ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
26	24, 13, 25	syl2anr 495	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
27		indistopon 20805	. . . 4 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )
28		iscn 21039	. . . 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 491	. . 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 301	. 2 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( J Cn { (/) , A } ) <-> f e. ( A ^m X ) ) )
31	30	eqrdv 2620	1 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J Cn { (/) , A } ) = ( A ^m X ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   {cpr 4179   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  indishmph  21601  indistgp  21904  indispconn  31216