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Theorem pwfseqlem2 8297
Description: Lemma for pwfseq 8302. (Contributed by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
pwfseqlem4.g  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
pwfseqlem4.x  |-  ( ph  ->  X  C_  A )
pwfseqlem4.h  |-  ( ph  ->  H : om -1-1-onto-> X )
pwfseqlem4.ps  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
pwfseqlem4.k  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
pwfseqlem4.d  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
pwfseqlem4.f  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
Assertion
Ref Expression
pwfseqlem2  |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `
 ( card `  Y
) ) )
Distinct variable groups:    n, r, w, x, z    D, n, z    w, G    w, K    H, r, x, z    ph, n, r, x, z    ps, n, z    A, n, r, x, z
Allowed substitution hints:    ph( w)    ps( x, w, r)    A( w)    D( x, w, r)    R( x, z, w, n, r)    F( x, z, w, n, r)    G( x, z, n, r)    H( w, n)    K( x, z, n, r)    X( x, z, w, n, r)    Y( x, z, w, n, r)

Proof of Theorem pwfseqlem2
Dummy variables  a 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . . 3  |-  ( a  =  Y  ->  (
a F s )  =  ( Y F s ) )
2 fveq2 5541 . . . 4  |-  ( a  =  Y  ->  ( card `  a )  =  ( card `  Y
) )
32fveq2d 5545 . . 3  |-  ( a  =  Y  ->  ( H `  ( card `  a ) )  =  ( H `  ( card `  Y ) ) )
41, 3eqeq12d 2310 . 2  |-  ( a  =  Y  ->  (
( a F s )  =  ( H `
 ( card `  a
) )  <->  ( Y F s )  =  ( H `  ( card `  Y ) ) ) )
5 oveq2 5882 . . 3  |-  ( s  =  R  ->  ( Y F s )  =  ( Y F R ) )
65eqeq1d 2304 . 2  |-  ( s  =  R  ->  (
( Y F s )  =  ( H `
 ( card `  Y
) )  <->  ( Y F R )  =  ( H `  ( card `  Y ) ) ) )
7 nfcv 2432 . . 3  |-  F/_ x
a
8 nfcv 2432 . . 3  |-  F/_ r
a
9 nfcv 2432 . . 3  |-  F/_ r
s
10 pwfseqlem4.f . . . . . 6  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
11 nfmpt21 5930 . . . . . 6  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
1210, 11nfcxfr 2429 . . . . 5  |-  F/_ x F
13 nfcv 2432 . . . . 5  |-  F/_ x
r
147, 12, 13nfov 5897 . . . 4  |-  F/_ x
( a F r )
1514nfeq1 2441 . . 3  |-  F/ x
( a F r )  =  ( H `
 ( card `  a
) )
16 nfmpt22 5931 . . . . . 6  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
1710, 16nfcxfr 2429 . . . . 5  |-  F/_ r F
188, 17, 9nfov 5897 . . . 4  |-  F/_ r
( a F s )
1918nfeq1 2441 . . 3  |-  F/ r ( a F s )  =  ( H `
 ( card `  a
) )
20 oveq1 5881 . . . 4  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
21 fveq2 5541 . . . . 5  |-  ( x  =  a  ->  ( card `  x )  =  ( card `  a
) )
2221fveq2d 5545 . . . 4  |-  ( x  =  a  ->  ( H `  ( card `  x ) )  =  ( H `  ( card `  a ) ) )
2320, 22eqeq12d 2310 . . 3  |-  ( x  =  a  ->  (
( x F r )  =  ( H `
 ( card `  x
) )  <->  ( a F r )  =  ( H `  ( card `  a ) ) ) )
24 oveq2 5882 . . . 4  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
2524eqeq1d 2304 . . 3  |-  ( r  =  s  ->  (
( a F r )  =  ( H `
 ( card `  a
) )  <->  ( a F s )  =  ( H `  ( card `  a ) ) ) )
26 vex 2804 . . . . . 6  |-  x  e. 
_V
27 vex 2804 . . . . . 6  |-  r  e. 
_V
28 fvex 5555 . . . . . . 7  |-  ( H `
 ( card `  x
) )  e.  _V
29 fvex 5555 . . . . . . 7  |-  ( D `
 |^| { z  e. 
om  |  -.  ( D `  z )  e.  x } )  e. 
_V
3028, 29ifex 3636 . . . . . 6  |-  if ( x  e.  Fin , 
( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  e. 
_V
3110ovmpt4g 5986 . . . . . 6  |-  ( ( x  e.  _V  /\  r  e.  _V  /\  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  e. 
_V )  ->  (
x F r )  =  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z
)  e.  x }
) ) )
3226, 27, 30, 31mp3an 1277 . . . . 5  |-  ( x F r )  =  if ( x  e. 
Fin ,  ( H `  ( card `  x
) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )
33 iftrue 3584 . . . . 5  |-  ( x  e.  Fin  ->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  =  ( H `  ( card `  x ) ) )
3432, 33syl5eq 2340 . . . 4  |-  ( x  e.  Fin  ->  (
x F r )  =  ( H `  ( card `  x )
) )
3534adantr 451 . . 3  |-  ( ( x  e.  Fin  /\  r  e.  _V )  ->  ( x F r )  =  ( H `
 ( card `  x
) ) )
367, 8, 9, 15, 19, 23, 25, 35vtocl2gaf 2863 . 2  |-  ( ( a  e.  Fin  /\  s  e.  _V )  ->  ( a F s )  =  ( H `
 ( card `  a
) ) )
374, 6, 36vtocl2ga 2864 1  |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `
 ( card `  Y
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ifcif 3578   ~Pcpw 3638   |^|cint 3878   U_ciun 3921   class class class wbr 4039    We wwe 4367   omcom 4672    X. cxp 4703   `'ccnv 4704   ran crn 4706   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788    ~<_ cdom 6877   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  pwfseqlem4a  8299  pwfseqlem4  8300
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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