Theorem cnfval 14917

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 14911	. . 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 3904	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = U. K )
5		toponuni 14738	. . . . . 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 3904	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = U. J )
10		toponuni 14738	. . . . . 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 6035	. . 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 2746	. . 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 2797	. 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 14737	. . 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 14737	. . 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 6823	. . . 4 |- ^m Fn ( _V X. _V )
22		toponmax 14748	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	22	elexd 2816	. . . . 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 14748	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
26	25	elexd 2816	. . . . 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 6050	. . . 4 |- ( ( ^m Fn ( _V X. _V ) /\ Y e. _V /\ X e. _V ) -> ( Y ^m X ) e. _V )
29	21, 24, 27, 28	mp3an2i 1378	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( Y ^m X ) e. _V )
30		rabexg 4233	. . 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 6148	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 1397   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   U.cuni 3893   X. cxp 4723   `'ccnv 4724   "cima 4728   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   ^m cmap 6816   Topctop 14720  TopOnctopon 14733   Cn ccn 14908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cn 14911

This theorem is referenced by:  cnovex  14919  iscn  14920