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Theorem domrancur1b 25303
Description: The currying of a mapping  F whose domain is  ( A  X.  B ) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 28-Apr-2010.)
Hypotheses
Ref Expression
domrancur1b.1  |-  A  e.  C
domrancur1b.2  |-  B  e.  D
domrancur1b.3  |-  B  =/=  (/)
domrancur1b.4  |-  F  Fn  ( A  X.  B
)
Assertion
Ref Expression
domrancur1b  |-  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
Distinct variable groups:    B, f    f, F
Allowed substitution hints:    A( f)    C( f)    D( f)

Proof of Theorem domrancur1b
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 domrancur1b.4 . . 3  |-  F  Fn  ( A  X.  B
)
2 domrancur1b.3 . . 3  |-  B  =/=  (/)
3 domrancur1b.1 . . 3  |-  A  e.  C
4 domrancur1b.2 . . 3  |-  B  e.  D
5 cur1vald 25302 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
61, 2, 3, 4, 5mp4an 654 . 2  |-  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
7 fnfun 5357 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
81, 7ax-mp 8 . . . . . 6  |-  Fun  F
9 2ndconst 6224 . . . . . . 7  |-  ( x  e.  A  ->  ( 2nd  |`  ( { x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V )
10 dff1o3 5494 . . . . . . . 8  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V  <->  ( ( 2nd  |`  ( { x }  X.  _V ) ) : ( { x }  X.  _V ) -onto-> _V  /\  Fun  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )
1110simprbi 450 . . . . . . 7  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V  ->  Fun  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )
129, 11syl 15 . . . . . 6  |-  ( x  e.  A  ->  Fun  `' ( 2nd  |`  ( { x }  X.  _V ) ) )
13 funco 5308 . . . . . 6  |-  ( ( Fun  F  /\  Fun  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
148, 12, 13sylancr 644 . . . . 5  |-  ( x  e.  A  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
15 dmco 5197 . . . . . 6  |-  dom  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )
16 imacnvcnv 5153 . . . . . . 7  |-  ( `' `' ( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )  =  ( ( 2nd  |`  ( {
x }  X.  _V ) ) " dom  F )
17 fndm 5359 . . . . . . . . . 10  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
181, 17ax-mp 8 . . . . . . . . 9  |-  dom  F  =  ( A  X.  B )
1918imaeq2i 5026 . . . . . . . 8  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) ) " dom  F )  =  ( ( 2nd  |`  ( {
x }  X.  _V ) ) " ( A  X.  B ) )
20 df-ima 4718 . . . . . . . . 9  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) ) " ( A  X.  B ) )  =  ran  ( ( 2nd  |`  ( {
x }  X.  _V ) )  |`  ( A  X.  B ) )
21 resres 4984 . . . . . . . . . . 11  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) )  |`  ( A  X.  B ) )  =  ( 2nd  |`  (
( { x }  X.  _V )  i^i  ( A  X.  B ) ) )
2221rneqi 4921 . . . . . . . . . 10  |-  ran  (
( 2nd  |`  ( { x }  X.  _V ) )  |`  ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B
) ) )
23 inxp 4834 . . . . . . . . . . . . . 14  |-  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { x }  i^i  A
)  X.  ( _V 
i^i  B ) )
24 snssi 3775 . . . . . . . . . . . . . . . 16  |-  ( x  e.  A  ->  { x }  C_  A )
25 df-ss 3179 . . . . . . . . . . . . . . . 16  |-  ( { x }  C_  A  <->  ( { x }  i^i  A )  =  { x } )
2624, 25sylib 188 . . . . . . . . . . . . . . 15  |-  ( x  e.  A  ->  ( { x }  i^i  A )  =  { x } )
27 incom 3374 . . . . . . . . . . . . . . . . 17  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
28 inv1 3494 . . . . . . . . . . . . . . . . 17  |-  ( B  i^i  _V )  =  B
2927, 28eqtri 2316 . . . . . . . . . . . . . . . 16  |-  ( _V 
i^i  B )  =  B
3029a1i 10 . . . . . . . . . . . . . . 15  |-  ( x  e.  A  ->  ( _V  i^i  B )  =  B )
3126, 30xpeq12d 4730 . . . . . . . . . . . . . 14  |-  ( x  e.  A  ->  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) )  =  ( { x }  X.  B ) )
3223, 31syl5eq 2340 . . . . . . . . . . . . 13  |-  ( x  e.  A  ->  (
( { x }  X.  _V )  i^i  ( A  X.  B ) )  =  ( { x }  X.  B ) )
3332reseq2d 4971 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( { x }  X.  B ) ) )
3433rneqd 4922 . . . . . . . . . . 11  |-  ( x  e.  A  ->  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( { x }  X.  B ) ) )
35 snnzg 3756 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  { x }  =/=  (/) )
36 fo2ndres 6160 . . . . . . . . . . . 12  |-  ( { x }  =/=  (/)  ->  ( 2nd  |`  ( { x }  X.  B ) ) : ( { x }  X.  B ) -onto-> B )
37 forn 5470 . . . . . . . . . . . 12  |-  ( ( 2nd  |`  ( {
x }  X.  B
) ) : ( { x }  X.  B ) -onto-> B  ->  ran  ( 2nd  |`  ( { x }  X.  B ) )  =  B )
3835, 36, 373syl 18 . . . . . . . . . . 11  |-  ( x  e.  A  ->  ran  ( 2nd  |`  ( {
x }  X.  B
) )  =  B )
3934, 38eqtrd 2328 . . . . . . . . . 10  |-  ( x  e.  A  ->  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
4022, 39syl5eq 2340 . . . . . . . . 9  |-  ( x  e.  A  ->  ran  ( ( 2nd  |`  ( { x }  X.  _V ) )  |`  ( A  X.  B ) )  =  B )
4120, 40syl5eq 2340 . . . . . . . 8  |-  ( x  e.  A  ->  (
( 2nd  |`  ( { x }  X.  _V ) ) " ( A  X.  B ) )  =  B )
4219, 41syl5eq 2340 . . . . . . 7  |-  ( x  e.  A  ->  (
( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )  =  B )
4316, 42syl5eq 2340 . . . . . 6  |-  ( x  e.  A  ->  ( `' `' ( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )  =  B )
4415, 43syl5eq 2340 . . . . 5  |-  ( x  e.  A  ->  dom  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  =  B )
45 df-fn 5274 . . . . 5  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  =  B ) )
4614, 44, 45sylanbrc 645 . . . 4  |-  ( x  e.  A  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  Fn  B )
47 rncoss 4961 . . . . 5  |-  ran  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  C_  ran  F
4847a1i 10 . . . 4  |-  ( x  e.  A  ->  ran  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  C_  ran  F )
49 df-f 5275 . . . 4  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F  <->  ( ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  Fn  B  /\  ran  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  C_  ran  F ) )
5046, 48, 49sylanbrc 645 . . 3  |-  ( x  e.  A  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) : B --> ran  F
)
51 elex 2809 . . . . 5  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F }  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e. 
_V )
5251adantl 452 . . . 4  |-  ( ( x  e.  A  /\  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } )  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e.  _V )
534elexi 2810 . . . . . . 7  |-  B  e. 
_V
54 xpexg 4816 . . . . . . . . . 10  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
553, 4, 54mp2an 653 . . . . . . . . 9  |-  ( A  X.  B )  e. 
_V
5618, 55eqeltri 2366 . . . . . . . 8  |-  dom  F  e.  _V
57 funrnex 5763 . . . . . . . 8  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
5856, 8, 57mp2 17 . . . . . . 7  |-  ran  F  e.  _V
5953, 58fpm 6816 . . . . . 6  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e.  ( ran  F  ^pm  B ) )
60 elex 2809 . . . . . 6  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e.  ( ran  F  ^pm  B )  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e. 
_V )
6159, 60syl 15 . . . . 5  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V )
6261adantl 452 . . . 4  |-  ( ( x  e.  A  /\  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V )
63 feq1 5391 . . . . 5  |-  ( f  =  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  -> 
( f : B --> ran  F  <->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) : B --> ran  F )
)
6463elabg 2928 . . . 4  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V  ->  ( ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e.  { f  |  f : B --> ran  F } 
<->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )
)
6552, 62, 64pm5.21nd 868 . . 3  |-  ( x  e.  A  ->  (
( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } 
<->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )
)
6650, 65mpbird 223 . 2  |-  ( x  e.  A  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e.  { f  |  f : B --> ran  F } )
676, 66fmpti 5699 1  |-  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137    ^pm cpm 6789   cur1ccur1 25297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-pm 6791  df-cur1 25299
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