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Theorem cnconst2 12238
Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( X X. { B } ) e. ( J Cn K ) )
Proof of Theorem cnconst2
Dummy variables x u y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 5277 . . 3 |- ( B e. Y -> ( X X. { B } ) : X --> Y )
213ad2ant3 985 . 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( X X. { B } ) : X --> Y )
32adantr 272 . . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> ( X X. { B } ) : X --> Y )
4 simpll3 1003 . . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> B e. Y )
5 simplr 502 . . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> x e. X )
6 fvconst2g 5586 . . . . . . . 8 |- ( ( B e. Y /\ x e. X ) -> ( ( X X. { B } ) ` x ) = B )
74, 5, 6syl2anc 406 . . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> ( ( X X. { B } ) ` x ) = B )
87eleq1d 2181 . . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> ( ( ( X X. { B } ) ` x ) e. y <-> B e. y ) )
9 simpll1 1001 . . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> J e. (TopOn ` X ) )
10 toponmax 12029 . . . . . . . . 9 |- ( J e. (TopOn ` X ) -> X e. J )
119, 10syl 14 . . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> X e. J )
12 simplr 502 . . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> x e. X )
13 df-ima 4510 . . . . . . . . 9 |- ( ( X X. { B } ) " X ) = ran ( ( X X. { B } ) |` X )
14 ssid 3081 . . . . . . . . . . . . 13 |- X C_ X
15 xpssres 4810 . . . . . . . . . . . . 13 |- ( X C_ X -> ( ( X X. { B } ) |` X ) = ( X X. { B } ) )
1614, 15ax-mp 7 . . . . . . . . . . . 12 |- ( ( X X. { B } ) |` X ) = ( X X. { B } )
1716rneqi 4725 . . . . . . . . . . 11 |- ran ( ( X X. { B } ) |` X ) = ran ( X X. { B } )
18 rnxpss 4926 . . . . . . . . . . 11 |- ran ( X X. { B } ) C_ { B }
1917, 18eqsstri 3093 . . . . . . . . . 10 |- ran ( ( X X. { B } ) |` X ) C_ { B }
20 simprr 504 . . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> B e. y )
2120snssd 3629 . . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> { B } C_ y )
2219, 21syl5ss 3072 . . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> ran ( ( X X. { B } ) |` X ) C_ y )
2313, 22syl5eqss 3107 . . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> ( ( X X. { B } ) " X ) C_ y )
24 eleq2 2176 . . . . . . . . . 10 |- ( u = X -> ( x e. u <-> x e. X ) )
25 imaeq2 4833 . . . . . . . . . . 11 |- ( u = X -> ( ( X X. { B } ) " u ) = ( ( X X. { B } ) " X ) )
2625sseq1d 3090 . . . . . . . . . 10 |- ( u = X -> ( ( ( X X. { B } ) " u ) C_ y <-> ( ( X X. { B } ) " X ) C_ y ) )
2724, 26anbi12d 462 . . . . . . . . 9 |- ( u = X -> ( ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) <-> ( x e. X /\ ( ( X X. { B } ) " X ) C_ y ) ) )
2827rspcev 2758 . . . . . . . 8 |- ( ( X e. J /\ ( x e. X /\ ( ( X X. { B } ) " X ) C_ y ) ) -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) )
2911, 12, 23, 28syl12anc 1195 . . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ ( y e. K /\ B e. y ) ) -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) )
3029expr 370 . . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> ( B e. y -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) ) )
318, 30sylbid 149 . . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) /\ y e. K ) -> ( ( ( X X. { B } ) ` x ) e. y -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) ) )
3231ralrimiva 2477 . . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> A. y e. K ( ( ( X X. { B } ) ` x ) e. y -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) ) )
33 simpl1 965 . . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> J e. (TopOn ` X ) )
34 simpl2 966 . . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> K e. (TopOn ` Y ) )
35 simpr 109 . . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> x e. X )
36 iscnp 12204 . . . . 5 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ x e. X ) -> ( ( X X. { B } ) e. ( ( J CnP K ) ` x ) <-> ( ( X X. { B } ) : X --> Y /\ A. y e. K ( ( ( X X. { B } ) ` x ) e. y -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) ) ) ) )
3733, 34, 35, 36syl3anc 1197 . . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> ( ( X X. { B } ) e. ( ( J CnP K ) ` x ) <-> ( ( X X. { B } ) : X --> Y /\ A. y e. K ( ( ( X X. { B } ) ` x ) e. y -> E. u e. J ( x e. u /\ ( ( X X. { B } ) " u ) C_ y ) ) ) ) )
383, 32, 37mpbir2and 909 . . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) /\ x e. X ) -> ( X X. { B } ) e. ( ( J CnP K ) ` x ) )
3938ralrimiva 2477 . 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> A. x e. X ( X X. { B } ) e. ( ( J CnP K ) ` x ) )
40 cncnp 12235 . . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( ( X X. { B } ) e. ( J Cn K ) <-> ( ( X X. { B } ) : X --> Y /\ A. x e. X ( X X. { B } ) e. ( ( J CnP K ) ` x ) ) ) )
41403adant3 982 . 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( ( X X. { B } ) e. ( J Cn K ) <-> ( ( X X. { B } ) : X --> Y /\ A. x e. X ( X X. { B } ) e. ( ( J CnP K ) ` x ) ) ) )
422, 39, 41mpbir2and 909 1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( X X. { B } ) e. ( J Cn K ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 943   = wceq 1312   e. wcel 1461   A.wral 2388   E.wrex 2389   C_ wss 3035   {csn 3491   X. cxp 4495   ran crn 4498   |` cres 4499   "cima 4500   -->wf 5075   ` cfv 5079  (class class class)co 5726  TopOnctopon 12014   Cn ccn 12191   CnP ccnp 12192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-map 6496  df-topgen 11978  df-top 12002  df-topon 12015  df-cn 12194  df-cnp 12195
This theorem is referenced by:  cnconst  12239  cnmptc  12287
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