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Theorem fpwwecbv 9011
Description: Lemma for fpwwe 9013. (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 455 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3516 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 459 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52sqxpeqd 5014 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3518 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 708 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4857 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4856 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 697 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 sneq 4026 . . . . . . . . . 10  |-  ( y  =  z  ->  { y }  =  { z } )
1211imaeq2d 5325 . . . . . . . . 9  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
1312fveq2d 5852 . . . . . . . 8  |-  ( y  =  z  ->  ( F `  ( `' r " { y } ) )  =  ( F `  ( `' r " { z } ) ) )
14 id 22 . . . . . . . 8  |-  ( y  =  z  ->  y  =  z )
1513, 14eqeq12d 2476 . . . . . . 7  |-  ( y  =  z  ->  (
( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' r " { z } ) )  =  z ) )
1615cbvralv 3081 . . . . . 6  |-  ( A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y  <->  A. z  e.  x  ( F `  ( `' r " { z } ) )  =  z )
174cnveqd 5167 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
1817imaeq1d 5324 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
1918fveq2d 5852 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( F `  ( `' r " {
z } ) )  =  ( F `  ( `' s " {
z } ) ) )
2019eqeq1d 2456 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( F `  ( `' r " {
z } ) )  =  z  <->  ( F `  ( `' s " { z } ) )  =  z ) )
212, 20raleqbidv 3065 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  ( F `  ( `' r " {
z } ) )  =  z  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2216, 21syl5bb 257 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2310, 22anbi12d 708 . . . 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 708 . . 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 4508 . 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 2483 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 367    = wceq 1398   A.wral 2804    C_ wss 3461   {csn 4016   {copab 4496    We wwe 4826    X. cxp 4986   `'ccnv 4987   "cima 4991   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578
This theorem is referenced by:  canthnum  9016  canthp1  9021
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