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Theorem fliftf 5814
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 447 . . . . 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 5808 . . . . . . . . . 10  |-  ( ph  ->  ( y F z  <->  E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
65exbidv 1612 . . . . . . . . 9  |-  ( ph  ->  ( E. z  y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
76adantr 451 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
8 rexcom4 2807 . . . . . . . . 9  |-  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) )
9 elisset 2798 . . . . . . . . . . . . . 14  |-  ( B  e.  S  ->  E. z 
z  =  B )
104, 9syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  E. z 
z  =  B )
1110biantrud 493 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
y  =  A  <->  ( y  =  A  /\  E. z 
z  =  B ) ) )
12 19.42v 1846 . . . . . . . . . . . 12  |-  ( E. z ( y  =  A  /\  z  =  B )  <->  ( y  =  A  /\  E. z 
z  =  B ) )
1311, 12syl6rbbr 255 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( E. z ( y  =  A  /\  z  =  B )  <->  y  =  A ) )
1413rexbidva 2560 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A ) )
1514adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  Fun  F )  ->  ( E. x  e.  X  E. z
( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
168, 15syl5bbr 250 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z E. x  e.  X  ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
177, 16bitrd 244 . . . . . . 7  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. x  e.  X  y  =  A ) )
1817abbidv 2397 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  { y  |  E. z  y F z }  =  {
y  |  E. x  e.  X  y  =  A } )
19 df-dm 4699 . . . . . 6  |-  dom  F  =  { y  |  E. z  y F z }
20 eqid 2283 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
2120rnmpt 4925 . . . . . 6  |-  ran  (
x  e.  X  |->  A )  =  { y  |  E. x  e.  X  y  =  A }
2218, 19, 213eqtr4g 2340 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  dom  F  =  ran  ( x  e.  X  |->  A ) )
23 df-fn 5258 . . . . 5  |-  ( F  Fn  ran  ( x  e.  X  |->  A )  <-> 
( Fun  F  /\  dom  F  =  ran  (
x  e.  X  |->  A ) ) )
241, 22, 23sylanbrc 645 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  F  Fn  ran  ( x  e.  X  |->  A ) )
252, 3, 4fliftrel 5807 . . . . . . 7  |-  ( ph  ->  F  C_  ( R  X.  S ) )
2625adantr 451 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  F  C_  ( R  X.  S ) )
27 rnss 4907 . . . . . 6  |-  ( F 
C_  ( R  X.  S )  ->  ran  F 
C_  ran  ( R  X.  S ) )
2826, 27syl 15 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  ran  ( R  X.  S
) )
29 rnxpss 5108 . . . . 5  |-  ran  ( R  X.  S )  C_  S
3028, 29syl6ss 3191 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  S
)
31 df-f 5259 . . . 4  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  <->  ( F  Fn  ran  ( x  e.  X  |->  A )  /\  ran  F 
C_  S ) )
3224, 30, 31sylanbrc 645 . . 3  |-  ( (
ph  /\  Fun  F )  ->  F : ran  ( x  e.  X  |->  A ) --> S )
3332ex 423 . 2  |-  ( ph  ->  ( Fun  F  ->  F : ran  ( x  e.  X  |->  A ) --> S ) )
34 ffun 5391 . 2  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  ->  Fun  F )
3533, 34impbid1 194 1  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  qliftf  6746  cygznlem2a  16521  pi1xfrf  18551  pi1cof  18557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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