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Theorem fseqdom 7899
Description: One half of fseqen 7900. (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 7590 . . 3  |-  om  e.  _V
2 ovex 6098 . . 3  |-  ( A  ^m  n )  e. 
_V
31, 2iunex 5983 . 2  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
4 xp1st 6368 . . . . . . . 8  |-  ( x  e.  ( om  X.  A )  ->  ( 1st `  x )  e. 
om )
5 peano2 4857 . . . . . . . 8  |-  ( ( 1st `  x )  e.  om  ->  suc  ( 1st `  x )  e.  om )
64, 5syl 16 . . . . . . 7  |-  ( x  e.  ( om  X.  A )  ->  suc  ( 1st `  x )  e.  om )
76adantl 453 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  suc  ( 1st `  x
)  e.  om )
8 xp2nd 6369 . . . . . . . . 9  |-  ( x  e.  ( om  X.  A )  ->  ( 2nd `  x )  e.  A )
98adantl 453 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( 2nd `  x
)  e.  A )
10 fconst6g 5624 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  A  ->  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } ) : suc  ( 1st `  x ) --> A )
119, 10syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A )
12 elmapg 7023 . . . . . . . 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 461 . . . . . . 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 224 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )
15 oveq2 6081 . . . . . . . 8  |-  ( n  =  suc  ( 1st `  x )  ->  ( A  ^m  n )  =  ( A  ^m  suc  ( 1st `  x ) ) )
1615eleq2d 2502 . . . . . . 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 3044 . . . . . 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 643 . . . . 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 4089 . . . . 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 204 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) )
2120ex 424 . . 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 4653 . . . . . . 7  |-  suc  ( 1st `  x )  =/=  (/)
23 fvex 5734 . . . . . . . 8  |-  ( 2nd `  x )  e.  _V
2423snnz 3914 . . . . . . 7  |-  { ( 2nd `  x ) }  =/=  (/)
25 xp11 5296 . . . . . . 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 654 . . . . . 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 6368 . . . . . . . . 9  |-  ( y  e.  ( om  X.  A )  ->  ( 1st `  y )  e. 
om )
28 peano4 4859 . . . . . . . . 9  |-  ( ( ( 1st `  x
)  e.  om  /\  ( 1st `  y )  e.  om )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
294, 27, 28syl2an 464 . . . . . . . 8  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
3029adantl 453 . . . . . . 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 3961 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  _V  ->  ( { ( 2nd `  x
) }  =  {
( 2nd `  y
) }  <->  ( 2nd `  x )  =  ( 2nd `  y ) ) )
3223, 31mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( { ( 2nd `  x ) }  =  { ( 2nd `  y ) }  <->  ( 2nd `  x
)  =  ( 2nd `  y ) ) )
3330, 32anbi12d 692 . . . . . 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 249 . . . . 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 6380 . . . . . 6  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( ( ( 1st `  x )  =  ( 1st `  y )  /\  ( 2nd `  x
)  =  ( 2nd `  y ) )  <->  x  =  y ) )
3635adantl 453 . . . . 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 245 . . . 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 424 . . 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 7140 . 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 17 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948   (/)c0 3620   {csn 3806   U_ciun 4085   class class class wbr 4204   suc csuc 4575   omcom 4837    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010    ~<_ cdom 7099
This theorem is referenced by:  fseqen  7900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-dom 7103
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