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Theorem fseqdom 7669
Description: One half of fseqen 7670. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
fseqdom  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Distinct variable group:    A, n
Allowed substitution hint:    V( n)

Proof of Theorem fseqdom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 7360 . . 3  |-  om  e.  _V
2 ovex 5899 . . 3  |-  ( A  ^m  n )  e. 
_V
31, 2iunex 5786 . 2  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
4 xp1st 6165 . . . . . . . 8  |-  ( x  e.  ( om  X.  A )  ->  ( 1st `  x )  e. 
om )
5 peano2 4692 . . . . . . . 8  |-  ( ( 1st `  x )  e.  om  ->  suc  ( 1st `  x )  e.  om )
64, 5syl 15 . . . . . . 7  |-  ( x  e.  ( om  X.  A )  ->  suc  ( 1st `  x )  e.  om )
76adantl 452 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  suc  ( 1st `  x
)  e.  om )
8 xp2nd 6166 . . . . . . . . 9  |-  ( x  e.  ( om  X.  A )  ->  ( 2nd `  x )  e.  A )
98adantl 452 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( 2nd `  x
)  e.  A )
10 fconst6g 5446 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  A  ->  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } ) : suc  ( 1st `  x ) --> A )
119, 10syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A )
12 elmapg 6801 . . . . . . . 8  |-  ( ( A  e.  V  /\  suc  ( 1st `  x
)  e.  om )  ->  ( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
136, 12sylan2 460 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
1411, 13mpbird 223 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )
15 oveq2 5882 . . . . . . . 8  |-  ( n  =  suc  ( 1st `  x )  ->  ( A  ^m  n )  =  ( A  ^m  suc  ( 1st `  x ) ) )
1615eleq2d 2363 . . . . . . 7  |-  ( n  =  suc  ( 1st `  x )  ->  (
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  n
)  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) ) )
1716rspcev 2897 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  e.  om  /\  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )  ->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
187, 14, 17syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  E. n  e.  om  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  n ) )
19 eliun 3925 . . . . 5  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  U_ n  e.  om  ( A  ^m  n )  <->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
2018, 19sylibr 203 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) )
2120ex 423 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( om 
X.  A )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) ) )
22 nsuceq0 4488 . . . . . . 7  |-  suc  ( 1st `  x )  =/=  (/)
23 fvex 5555 . . . . . . . 8  |-  ( 2nd `  x )  e.  _V
2423snnz 3757 . . . . . . 7  |-  { ( 2nd `  x ) }  =/=  (/)
25 xp11 5127 . . . . . . 7  |-  ( ( suc  ( 1st `  x
)  =/=  (/)  /\  {
( 2nd `  x
) }  =/=  (/) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  =  ( suc  ( 1st `  y )  X.  {
( 2nd `  y
) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) ) )
2622, 24, 25mp2an 653 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) )
27 xp1st 6165 . . . . . . . . 9  |-  ( y  e.  ( om  X.  A )  ->  ( 1st `  y )  e. 
om )
28 peano4 4694 . . . . . . . . 9  |-  ( ( ( 1st `  x
)  e.  om  /\  ( 1st `  y )  e.  om )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
294, 27, 28syl2an 463 . . . . . . . 8  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
3029adantl 452 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y
)  <->  ( 1st `  x
)  =  ( 1st `  y ) ) )
31 sneqbg 3799 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  _V  ->  ( { ( 2nd `  x
) }  =  {
( 2nd `  y
) }  <->  ( 2nd `  x )  =  ( 2nd `  y ) ) )
3223, 31mp1i 11 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( { ( 2nd `  x ) }  =  { ( 2nd `  y ) }  <->  ( 2nd `  x
)  =  ( 2nd `  y ) ) )
3330, 32anbi12d 691 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  =  suc  ( 1st `  y )  /\  { ( 2nd `  x
) }  =  {
( 2nd `  y
) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
3426, 33syl5bb 248 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
35 xpopth 6177 . . . . . 6  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( ( ( 1st `  x )  =  ( 1st `  y )  /\  ( 2nd `  x
)  =  ( 2nd `  y ) )  <->  x  =  y ) )
3635adantl 452 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) )  <->  x  =  y
) )
3734, 36bitrd 244 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) )
3837ex 423 . . 3  |-  ( A  e.  V  ->  (
( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) ) )
3921, 38dom2d 6918 . 2  |-  ( A  e.  V  ->  ( U_ n  e.  om  ( A  ^m  n
)  e.  _V  ->  ( om  X.  A )  ~<_ 
U_ n  e.  om  ( A  ^m  n
) ) )
403, 39mpi 16 1  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801   (/)c0 3468   {csn 3653   U_ciun 3921   class class class wbr 4039   suc csuc 4410   omcom 4672    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788    ~<_ cdom 6877
This theorem is referenced by:  fseqen  7670
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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-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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-dom 6881
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