Theorem cnfval 15005

Description: The set of all continuous functions from topology J to topology K. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnfval	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J Cn K ) = { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } )

Distinct variable groups:   y, f, K   f, X, y   f, Y, y   f, J, y

Proof of Theorem cnfval

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cn 14999	. . 3 |- Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } )
2	1	a1i 9	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } ) )
3		simprr 533	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> k = K )
4	3	unieqd 3909	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = U. K )
5		toponuni 14826	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 489	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> Y = U. K )
7	4, 6	eqtr4d 2267	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = Y )
8		simprl 531	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> j = J )
9	8	unieqd 3909	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = U. J )
10		toponuni 14826	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
11	10	ad2antrr 488	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> X = U. J )
12	9, 11	eqtr4d 2267	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = X )
13	7, 12	oveq12d 6046	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( U. k ^m U. j ) = ( Y ^m X ) )
14	8	eleq2d 2301	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( ( `' f " y ) e. j <-> ( `' f " y ) e. J ) )
15	3, 14	raleqbidv 2747	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( A. y e. k ( `' f " y ) e. j <-> A. y e. K ( `' f " y ) e. J ) )
16	13, 15	rabeqbidv 2798	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } = { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } )
17		topontop 14825	. . 3 |- ( J e. (TopOn ` X ) -> J e. Top )
18	17	adantr 276	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> J e. Top )
19		topontop 14825	. . 3 |- ( K e. (TopOn ` Y ) -> K e. Top )
20	19	adantl 277	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> K e. Top )
21		fnmap 6867	. . . 4 |- ^m Fn ( _V X. _V )
22		toponmax 14836	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	22	elexd 2817	. . . . 5 |- ( K e. (TopOn ` Y ) -> Y e. _V )
24	23	adantl 277	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> Y e. _V )
25		toponmax 14836	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
26	25	elexd 2817	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. _V )
27	26	adantr 276	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> X e. _V )
28		fnovex 6061	. . . 4 |- ( ( ^m Fn ( _V X. _V ) /\ Y e. _V /\ X e. _V ) -> ( Y ^m X ) e. _V )
29	21, 24, 27, 28	mp3an2i 1379	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( Y ^m X ) e. _V )
30		rabexg 4238	. . 3 |- ( ( Y ^m X ) e. _V -> { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } e. _V )
31	29, 30	syl 14	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } e. _V )
32	2, 16, 18, 20, 31	ovmpod 6159	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J Cn K ) = { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   _Vcvv 2803   U.cuni 3898   X. cxp 4729   `'ccnv 4730   "cima 4734   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   ^m cmap 6860   Topctop 14808  TopOnctopon 14821   Cn ccn 14996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-top 14809  df-topon 14822  df-cn 14999

This theorem is referenced by:  cnovex  15007  iscn  15008