Theorem cnindis 19960

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 4036	. . . . . . 7 |- ( x e. { (/) , A } -> ( x = (/) \/ x = A ) )
2		topontop 19594	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> J e. Top )
3	2	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> J e. Top )
4		0opn 19580	. . . . . . . . . 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 5321	. . . . . . . . . . 11 |- ( x = (/) -> ( `' f " x ) = ( `' f " (/) ) )
7		ima0 5340	. . . . . . . . . . 11 |- ( `' f " (/) ) = (/)
8	6, 7	syl6eq 2511	. . . . . . . . . 10 |- ( x = (/) -> ( `' f " x ) = (/) )
9	8	eleq1d 2523	. . . . . . . . 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 5995	. . . . . . . . . . 11 |- ( f : X --> A -> ( `' f " A ) = X )
12	11	adantl 464	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) = X )
13		toponmax 19596	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X e. J )
14	13	ad2antrr 723	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> X e. J )
15	12, 14	eqeltrd 2542	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> ( `' f " A ) e. J )
16		imaeq2 5321	. . . . . . . . . 10 |- ( x = A -> ( `' f " x ) = ( `' f " A ) )
17	16	eleq1d 2523	. . . . . . . . 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 378	. . . . . . 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 2866	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ A e. V ) /\ f : X --> A ) -> A. x e. { (/) , A } ( `' f " x ) e. J )
22	21	ex 432	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f : X --> A -> A. x e. { (/) , A } ( `' f " x ) e. J ) )
23	22	pm4.71d 632	. . 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 7425	. . . 4 |- ( ( A e. V /\ X e. J ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
26	24, 13, 25	syl2anr 476	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( f e. ( A ^m X ) <-> f : X --> A ) )
27		indistopon 19669	. . . 4 |- ( A e. V -> { (/) , A } e. (TopOn ` A ) )
28		iscn 19903	. . . 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 472	. . 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 2451	1 |- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J Cn { (/) , A } ) = ( A ^m X ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   (/)c0 3783   {cpr 4018   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   ^m cmap 7412   Topctop 19561  TopOnctopon 19562   Cn ccn 19892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19566  df-topon 19569  df-cn 19895

This theorem is referenced by:  indishmph  20465  indistgp  20765  indispcon  28943