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Theorem fliftf 6008
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 461 . . . . 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 6002 . . . . . . . . . 10  |-  ( ph  ->  ( y F z  <->  E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
65exbidv 1680 . . . . . . . . 9  |-  ( ph  ->  ( E. z  y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
76adantr 465 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
8 rexcom4 2992 . . . . . . . . 9  |-  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) )
9 elisset 2983 . . . . . . . . . . . . . 14  |-  ( B  e.  S  ->  E. z 
z  =  B )
104, 9syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  E. z 
z  =  B )
1110biantrud 507 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
y  =  A  <->  ( y  =  A  /\  E. z 
z  =  B ) ) )
12 19.42v 1924 . . . . . . . . . . . 12  |-  ( E. z ( y  =  A  /\  z  =  B )  <->  ( y  =  A  /\  E. z 
z  =  B ) )
1311, 12syl6rbbr 264 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( E. z ( y  =  A  /\  z  =  B )  <->  y  =  A ) )
1413rexbidva 2732 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A ) )
1514adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  Fun  F )  ->  ( E. x  e.  X  E. z
( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
168, 15syl5bbr 259 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z E. x  e.  X  ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
177, 16bitrd 253 . . . . . . 7  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. x  e.  X  y  =  A ) )
1817abbidv 2557 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  { y  |  E. z  y F z }  =  {
y  |  E. x  e.  X  y  =  A } )
19 df-dm 4850 . . . . . 6  |-  dom  F  =  { y  |  E. z  y F z }
20 eqid 2443 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
2120rnmpt 5085 . . . . . 6  |-  ran  (
x  e.  X  |->  A )  =  { y  |  E. x  e.  X  y  =  A }
2218, 19, 213eqtr4g 2500 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  dom  F  =  ran  ( x  e.  X  |->  A ) )
23 df-fn 5421 . . . . 5  |-  ( F  Fn  ran  ( x  e.  X  |->  A )  <-> 
( Fun  F  /\  dom  F  =  ran  (
x  e.  X  |->  A ) ) )
241, 22, 23sylanbrc 664 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  F  Fn  ran  ( x  e.  X  |->  A ) )
252, 3, 4fliftrel 6001 . . . . . . 7  |-  ( ph  ->  F  C_  ( R  X.  S ) )
2625adantr 465 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  F  C_  ( R  X.  S ) )
27 rnss 5068 . . . . . 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 5270 . . . . 5  |-  ran  ( R  X.  S )  C_  S
3028, 29syl6ss 3368 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  S
)
31 df-f 5422 . . . 4  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  <->  ( F  Fn  ran  ( x  e.  X  |->  A )  /\  ran  F 
C_  S ) )
3224, 30, 31sylanbrc 664 . . 3  |-  ( (
ph  /\  Fun  F )  ->  F : ran  ( x  e.  X  |->  A ) --> S )
3332ex 434 . 2  |-  ( ph  ->  ( Fun  F  ->  F : ran  ( x  e.  X  |->  A ) --> S ) )
34 ffun 5561 . 2  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  ->  Fun  F )
3533, 34impbid1 203 1  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   E.wrex 2716    C_ wss 3328   <.cop 3883   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ran crn 4841   Fun wfun 5412    Fn wfn 5413   -->wf 5414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426
This theorem is referenced by:  qliftf  7188  cygznlem2a  18000  pi1xfrf  20625  pi1cof  20631
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