Theorem cnfval 15218

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 15212	. . 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 537	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> k = K )
4	3	unieqd 3941	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = U. K )
5		toponuni 15039	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 493	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> Y = U. K )
7	4, 6	eqtr4d 2274	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = Y )
8		simprl 535	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> j = J )
9	8	unieqd 3941	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = U. J )
10		toponuni 15039	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
11	10	ad2antrr 492	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> X = U. J )
12	9, 11	eqtr4d 2274	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = X )
13	7, 12	oveq12d 6093	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( U. k ^m U. j ) = ( Y ^m X ) )
14	8	eleq2d 2308	. . . 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 2765	. . 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 2816	. 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 15038	. . 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 15038	. . 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 6919	. . . 4 |- ^m Fn ( _V X. _V )
22		toponmax 15049	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	22	elexd 2835	. . . . 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 15049	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
26	25	elexd 2835	. . . . 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 6108	. . . 4 |- ( ( ^m Fn ( _V X. _V ) /\ Y e. _V /\ X e. _V ) -> ( Y ^m X ) e. _V )
29	21, 24, 27, 28	mp3an2i 1383	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( Y ^m X ) e. _V )
30		rabexg 4274	. . 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 6206	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 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   U.cuni 3930   X. cxp 4767   `'ccnv 4768   "cima 4772   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   ^m cmap 6912   Topctop 15021  TopOnctopon 15034   Cn ccn 15209

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-top 15022  df-topon 15035  df-cn 15212

This theorem is referenced by:  cnovex  15220  iscn  15221