Theorem cnindis 20308

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 3985	. . . . . . 7 |- ( x e. { (/) , A } -> ( x = (/) \/ x = A ) )
2		topontop 19941	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
3	2	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> J e. Top )
4		0opn 19934	. . . . . . . . . 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 5164	. . . . . . . . . . 11 |- ( x = (/) -> ( `' f " x ) = ( `' f " (/) ) )
7		ima0 5183	. . . . . . . . . . 11 |- ( `' f " (/) ) = (/)
8	6, 7	syl6eq 2501	. . . . . . . . . 10 |- ( x = (/) -> ( `' f " x ) = (/) )
9	8	eleq1d 2513	. . . . . . . . 9 |- ( x = (/) -> ( ( `' f " x ) e. J <-> (/) e. J ) )
10	5, 9	syl5ibrcom 226	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = (/) -> ( `' f " x ) e. J ) )
11		fimacnv 6012	. . . . . . . . . . 11 |- ( f : X --> A -> ( `' f " A ) = X )
12	11	adantl 468	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) = X )
13		toponmax 19943	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X e. J )
14	13	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> X e. J )
15	12, 14	eqeltrd 2529	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) e. J )
16		imaeq2 5164	. . . . . . . . . 10 |- ( x = A -> ( `' f " x ) = ( `' f " A ) )
17	16	eleq1d 2513	. . . . . . . . 9 |- ( x = A -> ( ( `' f " x ) e. J <-> ( `' f " A ) e. J ) )
18	15, 17	syl5ibrcom 226	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x = A -> ( `' f " x ) e. J ) )
19	10, 18	jaod 382	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( ( x = (/) \/ x = A ) -> ( `' f " x ) e. J ) )
20	1, 19	syl5 33	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( x e. { (/) , A } -> ( `' f " x ) e. J ) )
21	20	ralrimiv 2800	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> A. x e. { (/) , A } ( `' f " x ) e. J )
22	21	ex 436	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f : X --> A -> A. x e. { (/) , A } ( `' f " x ) e. J ) )
23	22	pm4.71d 640	. . 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 7485	. . . 4 |- ( ( A e. V /\ X e. J ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
26	24, 13, 25	syl2anr 481	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
27		indistopon 20016	. . . 4 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )
28		iscn 20251	. . . 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 477	. . 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 290	. 2 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( J Cn { (/) , A } ) <-> f e. ( A ^m X ) ) )
31	30	eqrdv 2449	1 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J Cn { (/) , A } ) = ( A ^m X ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   (/)c0 3731   {cpr 3970   `'ccnv 4833   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290   ^m cmap 7472   Topctop 19917  TopOnctopon 19918   Cn ccn 20240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-top 19921  df-topon 19923  df-cn 20243

This theorem is referenced by:  indishmph  20813  indistgp  21115  indispcon  29957