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Theorem fpwwe2cbv 8268
Description: Lemma for fpwwe2 8281. (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 3218 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 447 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52, 2xpeq12d 4730 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3220 . . . . 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 4398 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4397 . . . . . 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 4730 . . . . . . . . . . . 12  |-  ( u  =  v  ->  (
u  X.  u )  =  ( v  X.  v ) )
1312ineq2d 3383 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
r  i^i  ( u  X.  u ) )  =  ( r  i^i  (
v  X.  v ) ) )
1411, 13oveq12d 5892 . . . . . . . . . 10  |-  ( u  =  v  ->  (
u F ( r  i^i  ( u  X.  u ) ) )  =  ( v F ( r  i^i  (
v  X.  v ) ) ) )
1514eqeq1d 2304 . . . . . . . . 9  |-  ( u  =  v  ->  (
( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  y ) )
1615cbvsbcv 3033 . . . . . . . 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 3664 . . . . . . . . . . 11  |-  ( y  =  z  ->  { y }  =  { z } )
1817imaeq2d 5028 . . . . . . . . . 10  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
19 dfsbcq 3006 . . . . . . . . . 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 2305 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  z ) )
2221sbcbidv 3058 . . . . . . . . 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 2777 . . . . . 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 4873 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
2726imaeq1d 5027 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
28 dfsbcq 3006 . . . . . . . . 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 3382 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  i^i  (
v  X.  v ) )  =  ( s  i^i  ( v  X.  v ) ) )
3130oveq2d 5890 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  ( v F ( r  i^i  ( v  X.  v ) ) )  =  ( v F ( s  i^i  ( v  X.  v
) ) ) )
3231eqeq1d 2304 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <-> 
( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3332sbcbidv 3058 . . . . . . . 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 2761 . . . . . 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 4104 . 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 2316 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 1632   A.wral 2556   [.wsbc 3004    i^i cin 3164    C_ wss 3165   {csn 3653   {copab 4092    We wwe 4367    X. cxp 4703   `'ccnv 4704   "cima 4708  (class class class)co 5874
This theorem is referenced by:  fpwwe2lem12  8279  fpwwe2lem13  8280  canthwe  8289  pwfseqlem5  8301
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877
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