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Theorem fpwwecbv 9087
Description: Lemma for fpwwe 9089. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
fpwwe.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
Assertion
Ref Expression
fpwwecbv  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) }
Distinct variable groups:    r, a,
s, x, A    y,
a, z, F, r, s, x
Allowed substitution hints:    A( y, z)    W( x, y, z, s, r, a)

Proof of Theorem fpwwecbv
StepHypRef Expression
1 fpwwe.1 . 2  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
2 simpl 464 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3445 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 468 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52sqxpeqd 4865 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3447 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 725 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4828 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4827 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 714 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 sneq 3969 . . . . . . . . . 10  |-  ( y  =  z  ->  { y }  =  { z } )
1211imaeq2d 5174 . . . . . . . . 9  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
1312fveq2d 5883 . . . . . . . 8  |-  ( y  =  z  ->  ( F `  ( `' r " { y } ) )  =  ( F `  ( `' r " { z } ) ) )
14 id 22 . . . . . . . 8  |-  ( y  =  z  ->  y  =  z )
1513, 14eqeq12d 2486 . . . . . . 7  |-  ( y  =  z  ->  (
( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' r " { z } ) )  =  z ) )
1615cbvralv 3005 . . . . . 6  |-  ( A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y  <->  A. z  e.  x  ( F `  ( `' r " { z } ) )  =  z )
174cnveqd 5015 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
1817imaeq1d 5173 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
1918fveq2d 5883 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( F `  ( `' r " {
z } ) )  =  ( F `  ( `' s " {
z } ) ) )
2019eqeq1d 2473 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( F `  ( `' r " {
z } ) )  =  z  <->  ( F `  ( `' s " { z } ) )  =  z ) )
212, 20raleqbidv 2987 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  ( F `  ( `' r " {
z } ) )  =  z  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2216, 21syl5bb 265 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2310, 22anbi12d 725 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y )  <-> 
( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) )
247, 23anbi12d 725 . . 3  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) )  <->  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) ) )
2524cbvopabv 4465 . 2  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) ) }
261, 25eqtri 2493 1  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452   A.wral 2756    C_ wss 3390   {csn 3959   {copab 4453    We wwe 4797    X. cxp 4837   `'ccnv 4838   "cima 4842   ` cfv 5589
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  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-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fv 5597
This theorem is referenced by:  canthnum  9092  canthp1  9097
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