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Theorem fliftf 5976
Description: The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftf  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftf
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  Fun  F )
2 flift.1 . . . . . . . . . . 11  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
3 flift.2 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
4 flift.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
52, 3, 4fliftel 5970 . . . . . . . . . 10  |-  ( ph  ->  ( y F z  <->  E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
65exbidv 1633 . . . . . . . . 9  |-  ( ph  ->  ( E. z  y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
76adantr 452 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
8 rexcom4 2918 . . . . . . . . 9  |-  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) )
9 elisset 2909 . . . . . . . . . . . . . 14  |-  ( B  e.  S  ->  E. z 
z  =  B )
104, 9syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  E. z 
z  =  B )
1110biantrud 494 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
y  =  A  <->  ( y  =  A  /\  E. z 
z  =  B ) ) )
12 19.42v 1917 . . . . . . . . . . . 12  |-  ( E. z ( y  =  A  /\  z  =  B )  <->  ( y  =  A  /\  E. z 
z  =  B ) )
1311, 12syl6rbbr 256 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( E. z ( y  =  A  /\  z  =  B )  <->  y  =  A ) )
1413rexbidva 2666 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A ) )
1514adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  Fun  F )  ->  ( E. x  e.  X  E. z
( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
168, 15syl5bbr 251 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z E. x  e.  X  ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
177, 16bitrd 245 . . . . . . 7  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. x  e.  X  y  =  A ) )
1817abbidv 2501 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  { y  |  E. z  y F z }  =  {
y  |  E. x  e.  X  y  =  A } )
19 df-dm 4828 . . . . . 6  |-  dom  F  =  { y  |  E. z  y F z }
20 eqid 2387 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
2120rnmpt 5056 . . . . . 6  |-  ran  (
x  e.  X  |->  A )  =  { y  |  E. x  e.  X  y  =  A }
2218, 19, 213eqtr4g 2444 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  dom  F  =  ran  ( x  e.  X  |->  A ) )
23 df-fn 5397 . . . . 5  |-  ( F  Fn  ran  ( x  e.  X  |->  A )  <-> 
( Fun  F  /\  dom  F  =  ran  (
x  e.  X  |->  A ) ) )
241, 22, 23sylanbrc 646 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  F  Fn  ran  ( x  e.  X  |->  A ) )
252, 3, 4fliftrel 5969 . . . . . . 7  |-  ( ph  ->  F  C_  ( R  X.  S ) )
2625adantr 452 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  F  C_  ( R  X.  S ) )
27 rnss 5038 . . . . . 6  |-  ( F 
C_  ( R  X.  S )  ->  ran  F 
C_  ran  ( R  X.  S ) )
2826, 27syl 16 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  ran  ( R  X.  S
) )
29 rnxpss 5241 . . . . 5  |-  ran  ( R  X.  S )  C_  S
3028, 29syl6ss 3303 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  S
)
31 df-f 5398 . . . 4  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  <->  ( F  Fn  ran  ( x  e.  X  |->  A )  /\  ran  F 
C_  S ) )
3224, 30, 31sylanbrc 646 . . 3  |-  ( (
ph  /\  Fun  F )  ->  F : ran  ( x  e.  X  |->  A ) --> S )
3332ex 424 . 2  |-  ( ph  ->  ( Fun  F  ->  F : ran  ( x  e.  X  |->  A ) --> S ) )
34 ffun 5533 . 2  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  ->  Fun  F )
3533, 34impbid1 195 1  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   E.wrex 2650    C_ wss 3263   <.cop 3760   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   dom cdm 4818   ran crn 4819   Fun wfun 5388    Fn wfn 5389   -->wf 5390
This theorem is referenced by:  qliftf  6928  cygznlem2a  16771  pi1xfrf  18949  pi1cof  18955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402
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