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Theorem fliftf 6226
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 468 . . . . 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 6220 . . . . . . . . . 10  |-  ( ph  ->  ( y F z  <->  E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
65exbidv 1776 . . . . . . . . 9  |-  ( ph  ->  ( E. z  y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
76adantr 472 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) ) )
8 rexcom4 3053 . . . . . . . . 9  |-  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. z E. x  e.  X  ( y  =  A  /\  z  =  B ) )
9 elisset 3043 . . . . . . . . . . . . . 14  |-  ( B  e.  S  ->  E. z 
z  =  B )
104, 9syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  E. z 
z  =  B )
1110biantrud 515 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
y  =  A  <->  ( y  =  A  /\  E. z 
z  =  B ) ) )
12 19.42v 1842 . . . . . . . . . . . 12  |-  ( E. z ( y  =  A  /\  z  =  B )  <->  ( y  =  A  /\  E. z 
z  =  B ) )
1311, 12syl6rbbr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( E. z ( y  =  A  /\  z  =  B )  <->  y  =  A ) )
1413rexbidva 2889 . . . . . . . . . 10  |-  ( ph  ->  ( E. x  e.  X  E. z ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A ) )
1514adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  Fun  F )  ->  ( E. x  e.  X  E. z
( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
168, 15syl5bbr 267 . . . . . . . 8  |-  ( (
ph  /\  Fun  F )  ->  ( E. z E. x  e.  X  ( y  =  A  /\  z  =  B )  <->  E. x  e.  X  y  =  A )
)
177, 16bitrd 261 . . . . . . 7  |-  ( (
ph  /\  Fun  F )  ->  ( E. z 
y F z  <->  E. x  e.  X  y  =  A ) )
1817abbidv 2589 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  { y  |  E. z  y F z }  =  {
y  |  E. x  e.  X  y  =  A } )
19 df-dm 4849 . . . . . 6  |-  dom  F  =  { y  |  E. z  y F z }
20 eqid 2471 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
2120rnmpt 5086 . . . . . 6  |-  ran  (
x  e.  X  |->  A )  =  { y  |  E. x  e.  X  y  =  A }
2218, 19, 213eqtr4g 2530 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  dom  F  =  ran  ( x  e.  X  |->  A ) )
23 df-fn 5592 . . . . 5  |-  ( F  Fn  ran  ( x  e.  X  |->  A )  <-> 
( Fun  F  /\  dom  F  =  ran  (
x  e.  X  |->  A ) ) )
241, 22, 23sylanbrc 677 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  F  Fn  ran  ( x  e.  X  |->  A ) )
252, 3, 4fliftrel 6219 . . . . . . 7  |-  ( ph  ->  F  C_  ( R  X.  S ) )
2625adantr 472 . . . . . 6  |-  ( (
ph  /\  Fun  F )  ->  F  C_  ( R  X.  S ) )
27 rnss 5069 . . . . . 6  |-  ( F 
C_  ( R  X.  S )  ->  ran  F 
C_  ran  ( R  X.  S ) )
2826, 27syl 17 . . . . 5  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  ran  ( R  X.  S
) )
29 rnxpss 5275 . . . . 5  |-  ran  ( R  X.  S )  C_  S
3028, 29syl6ss 3430 . . . 4  |-  ( (
ph  /\  Fun  F )  ->  ran  F  C_  S
)
31 df-f 5593 . . . 4  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  <->  ( F  Fn  ran  ( x  e.  X  |->  A )  /\  ran  F 
C_  S ) )
3224, 30, 31sylanbrc 677 . . 3  |-  ( (
ph  /\  Fun  F )  ->  F : ran  ( x  e.  X  |->  A ) --> S )
3332ex 441 . 2  |-  ( ph  ->  ( Fun  F  ->  F : ran  ( x  e.  X  |->  A ) --> S ) )
34 ffun 5742 . 2  |-  ( F : ran  ( x  e.  X  |->  A ) --> S  ->  Fun  F )
3533, 34impbid1 208 1  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   E.wrex 2757    C_ wss 3390   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840   Fun wfun 5583    Fn wfn 5584   -->wf 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597
This theorem is referenced by:  qliftf  7469  cygznlem2a  19215  pi1xfrf  22162  pi1cof  22168
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