Theorem cnfval 12988

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 12982	. . 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 527	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> k = K )
4	3	unieqd 3807	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = U. K )
5		toponuni 12807	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y = U. K )
6	5	ad2antlr 486	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> Y = U. K )
7	4, 6	eqtr4d 2206	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. k = Y )
8		simprl 526	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> j = J )
9	8	unieqd 3807	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = U. J )
10		toponuni 12807	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
11	10	ad2antrr 485	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> X = U. J )
12	9, 11	eqtr4d 2206	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> U. j = X )
13	7, 12	oveq12d 5871	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( j = J /\ k = K ) ) -> ( U. k ^m U. j ) = ( Y ^m X ) )
14	8	eleq2d 2240	. . . 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 2677	. . 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 2725	. 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 12806	. . 3 |- ( J e. (TopOn ` X ) -> J e. Top )
18	17	adantr 274	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> J e. Top )
19		topontop 12806	. . 3 |- ( K e. (TopOn ` Y ) -> K e. Top )
20	19	adantl 275	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> K e. Top )
21		fnmap 6633	. . . 4 |- ^m Fn ( _V X. _V )
22		toponmax 12817	. . . . . 6 |- ( K e. (TopOn ` Y ) -> Y e. K )
23	22	elexd 2743	. . . . 5 |- ( K e. (TopOn ` Y ) -> Y e. _V )
24	23	adantl 275	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> Y e. _V )
25		toponmax 12817	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
26	25	elexd 2743	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. _V )
27	26	adantr 274	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> X e. _V )
28		fnovex 5886	. . . 4 |- ( ( ^m Fn ( _V X. _V ) /\ Y e. _V /\ X e. _V ) -> ( Y ^m X ) e. _V )
29	21, 24, 27, 28	mp3an2i 1337	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( Y ^m X ) e. _V )
30		rabexg 4132	. . 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 5980	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 103   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730   U.cuni 3796   X. cxp 4609   `'ccnv 4610   "cima 4614   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   ^m cmap 6626   Topctop 12789  TopOnctopon 12802   Cn ccn 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cn 12982

This theorem is referenced by:  cnovex  12990  iscn  12991