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Theorem findcard 6604
Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
findcard.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
findcard.2  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
findcard.3  |-  ( x  =  y  ->  ( ph 
<->  th ) )
findcard.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
findcard.5  |-  ps
findcard.6  |-  ( y  e.  Fin  ->  ( A. z  e.  y  ch  ->  th ) )
Assertion
Ref Expression
findcard  |-  ( A  e.  Fin  ->  ta )
Distinct variable groups:    x, y, z, A    ps, x    ch, x    th, x    ta, x    ph, y, z
Allowed substitution hints:    ph( x)    ps( y,
z)    ch( y, z)    th( y,
z)    ta( y, z)

Proof of Theorem findcard
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 findcard.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2 isfi 6478 . . 3  |-  ( x  e.  Fin  <->  E. w  e.  om  x  ~~  w
)
3 breq2 3849 . . . . . . . 8  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
43imbi1d 229 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( x  ~~  w  ->  ph )  <->  ( x  ~~  (/) 
->  ph ) ) )
54albidv 1752 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  (/)  ->  ph )
) )
6 breq2 3849 . . . . . . . 8  |-  ( w  =  v  ->  (
x  ~~  w  <->  x  ~~  v ) )
76imbi1d 229 . . . . . . 7  |-  ( w  =  v  ->  (
( x  ~~  w  ->  ph )  <->  ( x  ~~  v  ->  ph )
) )
87albidv 1752 . . . . . 6  |-  ( w  =  v  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  v  ->  ph ) ) )
9 breq2 3849 . . . . . . . 8  |-  ( w  =  suc  v  -> 
( x  ~~  w  <->  x 
~~  suc  v )
)
109imbi1d 229 . . . . . . 7  |-  ( w  =  suc  v  -> 
( ( x  ~~  w  ->  ph )  <->  ( x  ~~  suc  v  ->  ph )
) )
1110albidv 1752 . . . . . 6  |-  ( w  =  suc  v  -> 
( A. x ( x  ~~  w  ->  ph )  <->  A. x ( x 
~~  suc  v  ->  ph ) ) )
12 en0 6512 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
13 findcard.5 . . . . . . . . 9  |-  ps
14 findcard.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
1513, 14mpbiri 166 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1612, 15sylbi 119 . . . . . . 7  |-  ( x 
~~  (/)  ->  ph )
1716ax-gen 1383 . . . . . 6  |-  A. x
( x  ~~  (/)  ->  ph )
18 peano2 4410 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  suc  v  e.  om )
19 breq2 3849 . . . . . . . . . . . . . 14  |-  ( w  =  suc  v  -> 
( y  ~~  w  <->  y 
~~  suc  v )
)
2019rspcev 2722 . . . . . . . . . . . . 13  |-  ( ( suc  v  e.  om  /\  y  ~~  suc  v
)  ->  E. w  e.  om  y  ~~  w
)
2118, 20sylan 277 . . . . . . . . . . . 12  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  E. w  e.  om  y  ~~  w )
22 isfi 6478 . . . . . . . . . . . 12  |-  ( y  e.  Fin  <->  E. w  e.  om  y  ~~  w
)
2321, 22sylibr 132 . . . . . . . . . . 11  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  y  e.  Fin )
24233adant2 962 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  -> 
y  e.  Fin )
25 dif1en 6595 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  om  /\  y  ~~  suc  v  /\  z  e.  y )  ->  ( y  \  {
z } )  ~~  v )
26253expa 1143 . . . . . . . . . . . . . . 15  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  (
y  \  { z } )  ~~  v
)
27 vex 2622 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
28 difexg 3980 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  _V  ->  (
y  \  { z } )  e.  _V )
2927, 28ax-mp 7 . . . . . . . . . . . . . . . 16  |-  ( y 
\  { z } )  e.  _V
30 breq1 3848 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( x  ~~  v 
<->  ( y  \  {
z } )  ~~  v ) )
31 findcard.2 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
3230, 31imbi12d 232 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  \  { z } )  ->  ( ( x 
~~  v  ->  ph )  <->  ( ( y  \  {
z } )  ~~  v  ->  ch ) ) )
3329, 32spcv 2712 . . . . . . . . . . . . . . 15  |-  ( A. x ( x  ~~  v  ->  ph )  ->  (
( y  \  {
z } )  ~~  v  ->  ch ) )
3426, 33syl5com 29 . . . . . . . . . . . . . 14  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  ( A. x ( x  ~~  v  ->  ph )  ->  ch ) )
3534ralrimdva 2453 . . . . . . . . . . . . 13  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  ( A. x
( x  ~~  v  ->  ph )  ->  A. z  e.  y  ch )
)
3635imp 122 . . . . . . . . . . . 12  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  A. x
( x  ~~  v  ->  ph ) )  ->  A. z  e.  y  ch )
3736an32s 535 . . . . . . . . . . 11  |-  ( ( ( v  e.  om  /\ 
A. x ( x 
~~  v  ->  ph )
)  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
38373impa 1138 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
39 findcard.6 . . . . . . . . . 10  |-  ( y  e.  Fin  ->  ( A. z  e.  y  ch  ->  th ) )
4024, 38, 39sylc 61 . . . . . . . . 9  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  th )
41403exp 1142 . . . . . . . 8  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  (
y  ~~  suc  v  ->  th ) ) )
4241alrimdv 1804 . . . . . . 7  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. y
( y  ~~  suc  v  ->  th ) ) )
43 breq1 3848 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  suc  v  <->  y  ~~  suc  v ) )
44 findcard.3 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  th ) )
4543, 44imbi12d 232 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  ~~  suc  v  ->  ph )  <->  ( y  ~~  suc  v  ->  th )
) )
4645cbvalv 1842 . . . . . . 7  |-  ( A. x ( x  ~~  suc  v  ->  ph )  <->  A. y ( y  ~~  suc  v  ->  th )
)
4742, 46syl6ibr 160 . . . . . 6  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. x
( x  ~~  suc  v  ->  ph ) ) )
485, 8, 11, 17, 47finds1 4417 . . . . 5  |-  ( w  e.  om  ->  A. x
( x  ~~  w  ->  ph ) )
494819.21bi 1495 . . . 4  |-  ( w  e.  om  ->  (
x  ~~  w  ->  ph ) )
5049rexlimiv 2483 . . 3  |-  ( E. w  e.  om  x  ~~  w  ->  ph )
512, 50sylbi 119 . 2  |-  ( x  e.  Fin  ->  ph )
521, 51vtoclga 2685 1  |-  ( A  e.  Fin  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619    \ cdif 2996   (/)c0 3286   {csn 3446   class class class wbr 3845   suc csuc 4192   omcom 4405    ~~ cen 6455   Fincfn 6457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-er 6292  df-en 6458  df-fin 6460
This theorem is referenced by:  xpfi  6640
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