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Theorem aceq3lem 8302
Description: Lemma for dfac3 8303. (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 2987 . . . . . 6  |-  y  e. 
_V
21rnex 6524 . . . . 5  |-  ran  y  e.  _V
32pwex 4487 . . . 4  |-  ~P ran  y  e.  _V
4 raleq 2929 . . . . 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 1680 . . . 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 3075 . . 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 4364 . . . . . . 7  |-  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
) )  =  { <. w ,  h >.  |  ( w  e.  dom  y  /\  h  =  ( f `  { u  |  w y u }
) ) }
97, 8eqtri 2463 . . . . . 6  |-  F  =  { <. w ,  h >.  |  ( w  e. 
dom  y  /\  h  =  ( f `  { u  |  w
y u } ) ) }
10 vex 2987 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
1110eldm 5049 . . . . . . . . . . . . . 14  |-  ( w  e.  dom  y  <->  E. u  w y u )
12 abn0 3668 . . . . . . . . . . . . . 14  |-  ( { u  |  w y u }  =/=  (/)  <->  E. u  w y u )
1311, 12bitr4i 252 . . . . . . . . . . . . 13  |-  ( w  e.  dom  y  <->  { u  |  w y u }  =/=  (/) )
14 vex 2987 . . . . . . . . . . . . . . . . 17  |-  u  e. 
_V
1510, 14brelrn 5082 . . . . . . . . . . . . . . . 16  |-  ( w y u  ->  u  e.  ran  y )
1615abssi 3439 . . . . . . . . . . . . . . 15  |-  { u  |  w y u }  C_ 
ran  y
172elpw2 4468 . . . . . . . . . . . . . . 15  |-  ( { u  |  w y u }  e.  ~P ran  y  <->  { u  |  w y u }  C_  ran  y )
1816, 17mpbir 209 . . . . . . . . . . . . . 14  |-  { u  |  w y u }  e.  ~P ran  y
19 neeq1 2628 . . . . . . . . . . . . . . . 16  |-  ( z  =  { u  |  w y u }  ->  ( z  =/=  (/)  <->  { u  |  w y u }  =/=  (/) ) )
20 fveq2 5703 . . . . . . . . . . . . . . . . 17  |-  ( z  =  { u  |  w y u }  ->  ( f `  z
)  =  ( f `
 { u  |  w y u }
) )
21 id 22 . . . . . . . . . . . . . . . . 17  |-  ( z  =  { u  |  w y u }  ->  z  =  { u  |  w y u }
)
2220, 21eleq12d 2511 . . . . . . . . . . . . . . . 16  |-  ( z  =  { u  |  w y u }  ->  ( ( f `  z )  e.  z  <-> 
( f `  {
u  |  w y u } )  e. 
{ u  |  w y u } ) )
2319, 22imbi12d 320 . . . . . . . . . . . . . . 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 3081 . . . . . . . . . . . . . 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 5 . . . . . . . . . . . . 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 217 . . . . . . . . . . . 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 429 . . . . . . . . . . 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 5713 . . . . . . . . . . . 12  |-  ( f `
 { u  |  w y u }
)  e.  _V
29 breq2 4308 . . . . . . . . . . . 12  |-  ( z  =  ( f `  { u  |  w
y u } )  ->  ( w y z  <->  w y ( f `  { u  |  w y u }
) ) )
30 breq2 4308 . . . . . . . . . . . . 13  |-  ( u  =  z  ->  (
w y u  <->  w y
z ) )
3130cbvabv 2570 . . . . . . . . . . . 12  |-  { u  |  w y u }  =  { z  |  w y z }
3228, 29, 31elab2 3121 . . . . . . . . . . 11  |-  ( ( f `  { u  |  w y u }
)  e.  { u  |  w y u }  <->  w y ( f `  { u  |  w
y u } ) )
3327, 32sylib 196 . . . . . . . . . 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 4308 . . . . . . . . . 10  |-  ( h  =  ( f `  { u  |  w
y u } )  ->  ( w y h  <->  w y ( f `  { u  |  w y u }
) ) )
3533, 34syl5ibrcom 222 . . . . . . . . 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 603 . . . . . . . 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 4629 . . . . . . 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 4365 . . . . . . 7  |-  { <. w ,  h >.  |  w y h }  C_  y
3937, 38syl6ss 3380 . . . . . 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 3412 . . . . 5  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  F  C_  y )
4128, 7fnmpti 5551 . . . . 5  |-  F  Fn  dom  y
421ssex 4448 . . . . . . 7  |-  ( F 
C_  y  ->  F  e.  _V )
4342adantr 465 . . . . . 6  |-  ( ( F  C_  y  /\  F  Fn  dom  y )  ->  F  e.  _V )
44 sseq1 3389 . . . . . . . 8  |-  ( g  =  F  ->  (
g  C_  y  <->  F  C_  y
) )
45 fneq1 5511 . . . . . . . 8  |-  ( g  =  F  ->  (
g  Fn  dom  y  <->  F  Fn  dom  y ) )
4644, 45anbi12d 710 . . . . . . 7  |-  ( g  =  F  ->  (
( g  C_  y  /\  g  Fn  dom  y )  <->  ( F  C_  y  /\  F  Fn  dom  y ) ) )
4746spcegv 3070 . . . . . 6  |-  ( F  e.  _V  ->  (
( F  C_  y  /\  F  Fn  dom  y )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) ) )
4843, 47mpcom 36 . . . . 5  |-  ( ( F  C_  y  /\  F  Fn  dom  y )  ->  E. g ( g 
C_  y  /\  g  Fn  dom  y ) )
4940, 41, 48sylancl 662 . . . 4  |-  ( A. z  e.  ~P  ran  y
( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  E. g
( g  C_  y  /\  g  Fn  dom  y ) )
5049exlimiv 1688 . . 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 16 . 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 3389 . . . 4  |-  ( g  =  f  ->  (
g  C_  y  <->  f  C_  y ) )
53 fneq1 5511 . . . 4  |-  ( g  =  f  ->  (
g  Fn  dom  y  <->  f  Fn  dom  y ) )
5452, 53anbi12d 710 . . 3  |-  ( g  =  f  ->  (
( g  C_  y  /\  g  Fn  dom  y )  <->  ( f  C_  y  /\  f  Fn 
dom  y ) ) )
5554cbvexv 1972 . 2  |-  ( E. g ( g  C_  y  /\  g  Fn  dom  y )  <->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
5651, 55sylib 196 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2618   A.wral 2727   _Vcvv 2984    C_ wss 3340   (/)c0 3649   ~Pcpw 3872   class class class wbr 4304   {copab 4361    e. cmpt 4362   dom cdm 4852   ran crn 4853    Fn wfn 5425   ` cfv 5430
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-fv 5438
This theorem is referenced by:  dfac3  8303
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