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Theorem fmpt2x 5768
Description: Functionality, domain and codomain of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.)
Hypothesis
Ref Expression
fmpt2x.1  F  ,  |->  C
Assertion
Ref Expression
fmpt2x  C  D  F : U_  { }  X.  --> D
Distinct variable groups:   ,,   ,   , D,
Allowed substitution hints:   ()    C(,)    F(,)

Proof of Theorem fmpt2x
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . 8 
_V
2 vex 2554 . . . . . . . 8 
_V
31, 2op1std 5717 . . . . . . 7  <. ,  >.  1st `
43csbeq1d 2852 . . . . . 6  <. ,  >.  [_ 1st `  ]_
[_ 2nd `  ]_ C  [_  ]_
[_ 2nd `  ]_ C
51, 2op2ndd 5718 . . . . . . . 8  <. ,  >.  2nd `
65csbeq1d 2852 . . . . . . 7  <. ,  >.  [_ 2nd `  ]_ C  [_  ]_ C
76csbeq2dv 2869 . . . . . 6  <. ,  >.  [_  ]_ [_ 2nd `  ]_ C  [_  ]_ [_  ]_ C
84, 7eqtrd 2069 . . . . 5  <. ,  >.  [_ 1st `  ]_
[_ 2nd `  ]_ C  [_  ]_
[_  ]_ C
98eleq1d 2103 . . . 4  <. ,  >.  [_ 1st `  ]_ [_ 2nd `  ]_ C  D  [_  ]_ [_  ]_ C  D
109raliunxp 4420 . . 3  U_  { }  X.  [_  ]_
[_ 1st `  ]_ [_ 2nd `  ]_ C  D  [_  ]_ [_  ]_ [_  ]_ C  D
11 nfv 1418 . . . . . . 7  F/  C
12 nfv 1418 . . . . . . 7  F/  C
13 nfv 1418 . . . . . . . . 9  F/
14 nfcsb1v 2876 . . . . . . . . . 10  F/_ [_  ]_
1514nfcri 2169 . . . . . . . . 9  F/  [_  ]_
1613, 15nfan 1454 . . . . . . . 8  F/  [_  ]_
17 nfcsb1v 2876 . . . . . . . . 9  F/_ [_  ]_ [_  ]_ C
1817nfeq2 2186 . . . . . . . 8  F/  [_  ]_ [_  ]_ C
1916, 18nfan 1454 . . . . . . 7  F/  [_  ]_  [_  ]_ [_  ]_ C
20 nfv 1418 . . . . . . . 8  F/  [_  ]_
21 nfcv 2175 . . . . . . . . . 10  F/_
22 nfcsb1v 2876 . . . . . . . . . 10  F/_ [_  ]_ C
2321, 22nfcsb 2878 . . . . . . . . 9  F/_ [_  ]_
[_  ]_ C
2423nfeq2 2186 . . . . . . . 8  F/  [_  ]_ [_  ]_ C
2520, 24nfan 1454 . . . . . . 7  F/ 
[_  ]_  [_  ]_
[_  ]_ C
26 eleq1 2097 . . . . . . . . . 10
2726adantr 261 . . . . . . . . 9
28 eleq1 2097 . . . . . . . . . 10
29 csbeq1a 2854 . . . . . . . . . . 11  [_  ]_
3029eleq2d 2104 . . . . . . . . . 10 
[_  ]_
3128, 30sylan9bbr 436 . . . . . . . . 9  [_  ]_
3227, 31anbi12d 442 . . . . . . . 8  [_  ]_
33 csbeq1a 2854 . . . . . . . . . 10  C  [_  ]_ C
34 csbeq1a 2854 . . . . . . . . . 10  [_  ]_ C 
[_  ]_
[_  ]_ C
3533, 34sylan9eqr 2091 . . . . . . . . 9  C  [_  ]_ [_  ]_ C
3635eqeq2d 2048 . . . . . . . 8  C  [_  ]_ [_  ]_ C
3732, 36anbi12d 442 . . . . . . 7  C  [_  ]_  [_  ]_ [_  ]_ C
3811, 12, 19, 25, 37cbvoprab12 5520 . . . . . 6  { <. <. ,  >. ,  >.  |  C }  { <. <. ,  >. ,  >.  |  [_  ]_  [_  ]_ [_  ]_ C }
39 df-mpt2 5460 . . . . . 6  ,  |->  C  { <. <. ,  >. ,  >.  |  C }
40 df-mpt2 5460 . . . . . 6  ,  [_  ]_  |->  [_  ]_
[_  ]_ C  { <. <. ,  >. ,  >.  |  [_  ]_  [_  ]_ [_  ]_ C }
4138, 39, 403eqtr4i 2067 . . . . 5  ,  |->  C  , 
[_  ]_  |->  [_  ]_
[_  ]_ C
42 fmpt2x.1 . . . . 5  F  ,  |->  C
438mpt2mptx 5537 . . . . 5  U_  { }  X.  [_  ]_ 
|->  [_ 1st `  ]_ [_ 2nd `  ]_ C  ,  [_  ]_  |->  [_  ]_ [_  ]_ C
4441, 42, 433eqtr4i 2067 . . . 4  F  U_  { }  X.  [_  ]_  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
4544fmpt 5262 . . 3  U_  { }  X.  [_  ]_
[_ 1st `  ]_ [_ 2nd `  ]_ C  D  F : U_  { }  X.  [_  ]_ --> D
4610, 45bitr3i 175 . 2  [_  ]_ [_  ]_ [_  ]_ C  D  F : U_  { }  X.  [_  ]_ --> D
47 nfv 1418 . . 3  F/  C  D
4817nfel1 2185 . . . 4  F/ [_  ]_ [_  ]_ C  D
4914, 48nfralxy 2354 . . 3  F/  [_  ]_ [_  ]_ [_  ]_ C  D
50 nfv 1418 . . . . 5  F/  C  D
5122nfel1 2185 . . . . 5  F/
[_  ]_ C  D
5233eleq1d 2103 . . . . 5  C  D  [_  ]_ C  D
5350, 51, 52cbvral 2523 . . . 4  C  D  [_  ]_ C  D
5434eleq1d 2103 . . . . 5  [_  ]_ C  D  [_  ]_ [_  ]_ C  D
5529, 54raleqbidv 2511 . . . 4  [_  ]_ C  D  [_  ]_ [_  ]_ [_  ]_ C  D
5653, 55syl5bb 181 . . 3  C  D  [_  ]_ [_  ]_ [_  ]_ C  D
5747, 49, 56cbvral 2523 . 2  C  D  [_  ]_ [_  ]_ [_  ]_ C  D
58 nfcv 2175 . . . 4  F/_ { }  X.
59 nfcv 2175 . . . . 5  F/_ { }
6059, 14nfxp 4314 . . . 4  F/_ { }  X.  [_  ]_
61 sneq 3378 . . . . 5  { }  { }
6261, 29xpeq12d 4313 . . . 4  { }  X.  { }  X.  [_  ]_
6358, 60, 62cbviun 3685 . . 3  U_  { }  X. 
U_  { }  X.  [_  ]_
6463feq2i 4983 . 2  F : U_  { }  X.  --> D  F : U_  { }  X.  [_  ]_ --> D
6546, 57, 643bitr4i 201 1  C  D  F : U_  { }  X.  --> D
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648    |-> cmpt 3809    X. cxp 4286   -->wf 4841   ` cfv 4845   {coprab 5456    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  fmpt2  5769
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