Theorem cnindis 19012

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 3995	. . . . . . 7 |- ( x e. { (/) , A } -> ( x = (/) \/ x = A ) )
2		topontop 18647	. . . . . . . . . . 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 18633	. . . . . . . . . 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 5263	. . . . . . . . . . 11 |- ( x = (/) -> ( `' f " x ) = ( `' f " (/) ) )
7		ima0 5282	. . . . . . . . . . 11 |- ( `' f " (/) ) = (/)
8	6, 7	syl6eq 2508	. . . . . . . . . 10 |- ( x = (/) -> ( `' f " x ) = (/) )
9	8	eleq1d 2520	. . . . . . . . 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 5934	. . . . . . . . . . 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 18649	. . . . . . . . . . 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 2539	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) e. J )
16		imaeq2 5263	. . . . . . . . . 10 |- ( x = A -> ( `' f " x ) = ( `' f " A ) )
17	16	eleq1d 2520	. . . . . . . . 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 2820	. . . . 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 7327	. . . 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 18721	. . . 4 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )
28		iscn 18955	. . . 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 2448	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 1370   e. wcel 1758   A.wral 2795   (/)c0 3735   {cpr 3977   `'ccnv 4937   "cima 4941   -->wf 5512   ` cfv 5516  (class class class)co 6190   ^m cmap 7314   Topctop 18614  TopOnctopon 18615   Cn ccn 18944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-top 18619  df-topon 18622  df-cn 18947

This theorem is referenced by:  indishmph  19487  indistgp  19787  indispcon  27257