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Theorem aceq3lem 7763
Description: Lemma for dfac3 7764. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
aceq3lem.1  |-  F  =  ( w  e.  dom  y  |->  ( f `  { u  |  w
y u } ) )
Assertion
Ref Expression
aceq3lem  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
Distinct variable group:    x, y, z, w, u, f
Allowed substitution hints:    F( x, y, z, w, u, f)

Proof of Theorem aceq3lem
Dummy variables  h  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . 6  |-  y  e. 
_V
21rnex 4958 . . . . 5  |-  ran  y  e.  _V
32pwex 4209 . . . 4  |-  ~P ran  y  e.  _V
4 raleq 2749 . . . . 5  |-  ( x  =  ~P ran  y  ->  ( A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  <->  A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z ) ) )
54exbidv 1616 . . . 4  |-  ( x  =  ~P ran  y  ->  ( E. f A. z  e.  x  (
z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  E. f A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z ) ) )
63, 5spcv 2887 . . 3  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. f A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
7 aceq3lem.1 . . . . . . 7  |-  F  =  ( w  e.  dom  y  |->  ( f `  { u  |  w
y u } ) )
8 df-mpt 4095 . . . . . . 7  |-  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
) )  =  { <. w ,  h >.  |  ( w  e.  dom  y  /\  h  =  ( f `  { u  |  w y u }
) ) }
97, 8eqtri 2316 . . . . . 6  |-  F  =  { <. w ,  h >.  |  ( w  e. 
dom  y  /\  h  =  ( f `  { u  |  w
y u } ) ) }
10 vex 2804 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
1110eldm 4892 . . . . . . . . . . . . . 14  |-  ( w  e.  dom  y  <->  E. u  w y u )
12 abn0 3486 . . . . . . . . . . . . . 14  |-  ( { u  |  w y u }  =/=  (/)  <->  E. u  w y u )
1311, 12bitr4i 243 . . . . . . . . . . . . 13  |-  ( w  e.  dom  y  <->  { u  |  w y u }  =/=  (/) )
14 vex 2804 . . . . . . . . . . . . . . . . 17  |-  u  e. 
_V
1510, 14brelrn 4925 . . . . . . . . . . . . . . . 16  |-  ( w y u  ->  u  e.  ran  y )
1615abssi 3261 . . . . . . . . . . . . . . 15  |-  { u  |  w y u }  C_ 
ran  y
172elpw2 4191 . . . . . . . . . . . . . . 15  |-  ( { u  |  w y u }  e.  ~P ran  y  <->  { u  |  w y u }  C_  ran  y )
1816, 17mpbir 200 . . . . . . . . . . . . . 14  |-  { u  |  w y u }  e.  ~P ran  y
19 neeq1 2467 . . . . . . . . . . . . . . . 16  |-  ( z  =  { u  |  w y u }  ->  ( z  =/=  (/)  <->  { u  |  w y u }  =/=  (/) ) )
20 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( z  =  { u  |  w y u }  ->  ( f `  z
)  =  ( f `
 { u  |  w y u }
) )
21 id 19 . . . . . . . . . . . . . . . . 17  |-  ( z  =  { u  |  w y u }  ->  z  =  { u  |  w y u }
)
2220, 21eleq12d 2364 . . . . . . . . . . . . . . . 16  |-  ( z  =  { u  |  w y u }  ->  ( ( f `  z )  e.  z  <-> 
( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } ) )
2319, 22imbi12d 311 . . . . . . . . . . . . . . 15  |-  ( z  =  { u  |  w y u }  ->  ( ( z  =/=  (/)  ->  ( f `  z )  e.  z )  <->  ( { u  |  w y u }  =/=  (/)  ->  ( f `  { u  |  w y u } )  e.  { u  |  w y u }
) ) )
2423rspcv 2893 . . . . . . . . . . . . . 14  |-  ( { u  |  w y u }  e.  ~P ran  y  ->  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  ( { u  |  w
y u }  =/=  (/) 
->  ( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } ) ) )
2518, 24ax-mp 8 . . . . . . . . . . . . 13  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  ( { u  |  w
y u }  =/=  (/) 
->  ( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } ) )
2613, 25syl5bi 208 . . . . . . . . . . . 12  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  (
w  e.  dom  y  ->  ( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } ) )
2726imp 418 . . . . . . . . . . 11  |-  ( ( A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z )  /\  w  e. 
dom  y )  -> 
( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } )
28 fvex 5555 . . . . . . . . . . . 12  |-  ( f `
 { u  |  w y u }
)  e.  _V
29 breq2 4043 . . . . . . . . . . . 12  |-  ( z  =  ( f `  { u  |  w
y u } )  ->  ( w y z  <->  w y ( f `  { u  |  w y u }
) ) )
30 breq2 4043 . . . . . . . . . . . . 13  |-  ( u  =  z  ->  (
w y u  <->  w y
z ) )
3130cbvabv 2415 . . . . . . . . . . . 12  |-  { u  |  w y u }  =  { z  |  w y z }
3228, 29, 31elab2 2930 . . . . . . . . . . 11  |-  ( ( f `  { u  |  w y u }
)  e.  { u  |  w y u }  <->  w y ( f `  { u  |  w
y u } ) )
3327, 32sylib 188 . . . . . . . . . 10  |-  ( ( A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z )  /\  w  e. 
dom  y )  ->  w y ( f `
 { u  |  w y u }
) )
34 breq2 4043 . . . . . . . . . 10  |-  ( h  =  ( f `  { u  |  w
y u } )  ->  ( w y h  <->  w y ( f `  { u  |  w y u }
) ) )
3533, 34syl5ibrcom 213 . . . . . . . . 9  |-  ( ( A. z  e.  ~P  ran  y ( z  =/=  (/)  ->  ( f `  z )  e.  z )  /\  w  e. 
dom  y )  -> 
( h  =  ( f `  { u  |  w y u }
)  ->  w y
h ) )
3635expimpd 586 . . . . . . . 8  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  (
( w  e.  dom  y  /\  h  =  ( f `  { u  |  w y u }
) )  ->  w
y h ) )
3736ssopab2dv 4309 . . . . . . 7  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  { <. w ,  h >.  |  ( w  e.  dom  y  /\  h  =  (
f `  { u  |  w y u }
) ) }  C_  {
<. w ,  h >.  |  w y h }
)
38 opabss 4096 . . . . . . 7  |-  { <. w ,  h >.  |  w y h }  C_  y
3937, 38syl6ss 3204 . . . . . 6  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  { <. w ,  h >.  |  ( w  e.  dom  y  /\  h  =  (
f `  { u  |  w y u }
) ) }  C_  y )
409, 39syl5eqss 3235 . . . . 5  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  F  C_  y )
4128, 7fnmpti 5388 . . . . 5  |-  F  Fn  dom  y
421ssex 4174 . . . . . . 7  |-  ( F 
C_  y  ->  F  e.  _V )
4342adantr 451 . . . . . 6  |-  ( ( F  C_  y  /\  F  Fn  dom  y )  ->  F  e.  _V )
44 sseq1 3212 . . . . . . . 8  |-  ( g  =  F  ->  (
g  C_  y  <->  F  C_  y
) )
45 fneq1 5349 . . . . . . . 8  |-  ( g  =  F  ->  (
g  Fn  dom  y  <->  F  Fn  dom  y ) )
4644, 45anbi12d 691 . . . . . . 7  |-  ( g  =  F  ->  (
( g  C_  y  /\  g  Fn  dom  y )  <->  ( F  C_  y  /\  F  Fn  dom  y ) ) )
4746spcegv 2882 . . . . . 6  |-  ( F  e.  _V  ->  (
( F  C_  y  /\  F  Fn  dom  y )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) ) )
4843, 47mpcom 32 . . . . 5  |-  ( ( F  C_  y  /\  F  Fn  dom  y )  ->  E. g ( g 
C_  y  /\  g  Fn  dom  y ) )
4940, 41, 48sylancl 643 . . . 4  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) )
5049exlimiv 1624 . . 3  |-  ( E. f A. z  e. 
~P  ran  y (
z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) )
516, 50syl 15 . 2  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) )
52 sseq1 3212 . . . 4  |-  ( g  =  f  ->  (
g  C_  y  <->  f  C_  y ) )
53 fneq1 5349 . . . 4  |-  ( g  =  f  ->  (
g  Fn  dom  y  <->  f  Fn  dom  y ) )
5452, 53anbi12d 691 . . 3  |-  ( g  =  f  ->  (
( g  C_  y  /\  g  Fn  dom  y )  <->  ( f  C_  y  /\  f  Fn 
dom  y ) ) )
5554cbvexv 1956 . 2  |-  ( E. g ( g  C_  y  /\  g  Fn  dom  y )  <->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
5651, 55sylib 188 1  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039   {copab 4092    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  dfac3  7764
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-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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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