Theorem cnfval 14514

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 14508	. . 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 531	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> k = K )
4	3	unieqd 3851	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = U. K )
5		toponuni 14335	. . . . . 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 2232	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = Y )
8		simprl 529	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> j = J )
9	8	unieqd 3851	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = U. J )
10		toponuni 14335	. . . . . 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 2232	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = X )
13	7, 12	oveq12d 5943	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( U. k ^m U. j ) = ( Y ^m X ) )
14	8	eleq2d 2266	. . . 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 2709	. . 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 2758	. 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 14334	. . 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 14334	. . 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 6723	. . . 4 |- ^m Fn ( _V X. _V )
22		toponmax 14345	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	22	elexd 2776	. . . . 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 14345	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
26	25	elexd 2776	. . . . 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 5958	. . . 4 |- ( ( ^m Fn ( _V X. _V ) /\ Y e. _V /\ X e. _V ) -> ( Y ^m X ) e. _V )
29	21, 24, 27, 28	mp3an2i 1353	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( Y ^m X ) e. _V )
30		rabexg 4177	. . 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 6054	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 1364   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763   U.cuni 3840   X. cxp 4662   `'ccnv 4663   "cima 4667   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   ^m cmap 6716   Topctop 14317  TopOnctopon 14330   Cn ccn 14505

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-top 14318  df-topon 14331  df-cn 14508

This theorem is referenced by:  cnovex  14516  iscn  14517