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Theorem fpwwe2cbv 8252
Description: Lemma for fpwwe2 8265. (Contributed by Mario Carneiro, 3-Jun-2015.)
Hypothesis
Ref Expression
fpwwe2.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
Assertion
Ref Expression
fpwwe2cbv  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
Distinct variable groups:    y, u    r, a, s, u, v, x, y, z, F    A, a, r, s, x, z
Allowed substitution hints:    A( y, v, u)    W( x, y, z, v, u, s, r, a)

Proof of Theorem fpwwe2cbv
StepHypRef Expression
1 fpwwe2.1 . 2  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
2 simpl 443 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3205 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 447 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52, 2xpeq12d 4714 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3207 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 691 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4382 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4381 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 680 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 id 19 . . . . . . . . . . 11  |-  ( u  =  v  ->  u  =  v )
1211, 11xpeq12d 4714 . . . . . . . . . . . 12  |-  ( u  =  v  ->  (
u  X.  u )  =  ( v  X.  v ) )
1312ineq2d 3370 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
r  i^i  ( u  X.  u ) )  =  ( r  i^i  (
v  X.  v ) ) )
1411, 13oveq12d 5876 . . . . . . . . . 10  |-  ( u  =  v  ->  (
u F ( r  i^i  ( u  X.  u ) ) )  =  ( v F ( r  i^i  (
v  X.  v ) ) ) )
1514eqeq1d 2291 . . . . . . . . 9  |-  ( u  =  v  ->  (
( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  y ) )
1615cbvsbcv 3020 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <->  [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y )
17 sneq 3651 . . . . . . . . . . 11  |-  ( y  =  z  ->  { y }  =  { z } )
1817imaeq2d 5012 . . . . . . . . . 10  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
19 dfsbcq 2993 . . . . . . . . . 10  |-  ( ( `' r " {
y } )  =  ( `' r " { z } )  ->  ( [. ( `' r " {
y } )  / 
v ]. ( v F ( r  i^i  (
v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y ) )
2018, 19syl 15 . . . . . . . . 9  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  y ) )
21 eqeq2 2292 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  z ) )
2221sbcbidv 3045 . . . . . . . . 9  |-  ( y  =  z  ->  ( [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2320, 22bitrd 244 . . . . . . . 8  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2416, 23syl5bb 248 . . . . . . 7  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2524cbvralv 2764 . . . . . 6  |-  ( A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <->  A. z  e.  x  [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  z )
264cnveqd 4857 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
2726imaeq1d 5011 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
28 dfsbcq 2993 . . . . . . . . 9  |-  ( ( `' r " {
z } )  =  ( `' s " { z } )  ->  ( [. ( `' r " {
z } )  / 
v ]. ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2927, 28syl 15 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
304ineq1d 3369 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  i^i  (
v  X.  v ) )  =  ( s  i^i  ( v  X.  v ) ) )
3130oveq2d 5874 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  ( v F ( r  i^i  ( v  X.  v ) ) )  =  ( v F ( s  i^i  ( v  X.  v
) ) ) )
3231eqeq1d 2291 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <-> 
( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3332sbcbidv 3045 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' s " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3429, 33bitrd 244 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
352, 34raleqbidv 2748 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3625, 35syl5bb 248 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3710, 36anbi12d 691 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y )  <-> 
( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) )
387, 37anbi12d 691 . . 3  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y ) )  <->  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) ) )
3938cbvopabv 4088 . 2  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v F ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }
401, 39eqtri 2303 1  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   [.wsbc 2991    i^i cin 3151    C_ wss 3152   {csn 3640   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   "cima 4692  (class class class)co 5858
This theorem is referenced by:  fpwwe2lem12  8263  fpwwe2lem13  8264  canthwe  8273  pwfseqlem5  8285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861
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