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Theorem fin23lem15 8753
Description: Lemma for fin23 8808. 
U is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
Assertion
Ref Expression
fin23lem15  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B )
)
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    B( u, t, i)    U( t)

Proof of Theorem fin23lem15
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . 3  |-  ( b  =  B  ->  ( U `  b )  =  ( U `  B ) )
21sseq1d 3488 . 2  |-  ( b  =  B  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  B )  C_  ( U `  B )
) )
3 fveq2 5872 . . 3  |-  ( b  =  a  ->  ( U `  b )  =  ( U `  a ) )
43sseq1d 3488 . 2  |-  ( b  =  a  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  a )  C_  ( U `  B )
) )
5 fveq2 5872 . . 3  |-  ( b  =  suc  a  -> 
( U `  b
)  =  ( U `
 suc  a )
)
65sseq1d 3488 . 2  |-  ( b  =  suc  a  -> 
( ( U `  b )  C_  ( U `  B )  <->  ( U `  suc  a
)  C_  ( U `  B ) ) )
7 fveq2 5872 . . 3  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
87sseq1d 3488 . 2  |-  ( b  =  A  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  A )  C_  ( U `  B )
) )
9 ssid 3480 . . 3  |-  ( U `
 B )  C_  ( U `  B )
109a1i 11 . 2  |-  ( B  e.  om  ->  ( U `  B )  C_  ( U `  B
) )
11 fin23lem.a . . . . 5  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
1211fin23lem13 8751 . . . 4  |-  ( a  e.  om  ->  ( U `  suc  a ) 
C_  ( U `  a ) )
1312ad2antrr 730 . . 3  |-  ( ( ( a  e.  om  /\  B  e.  om )  /\  B  C_  a )  ->  ( U `  suc  a )  C_  ( U `  a )
)
14 sstr2 3468 . . 3  |-  ( ( U `  suc  a
)  C_  ( U `  a )  ->  (
( U `  a
)  C_  ( U `  B )  ->  ( U `  suc  a ) 
C_  ( U `  B ) ) )
1513, 14syl 17 . 2  |-  ( ( ( a  e.  om  /\  B  e.  om )  /\  B  C_  a )  ->  ( ( U `
 a )  C_  ( U `  B )  ->  ( U `  suc  a )  C_  ( U `  B )
) )
162, 4, 6, 8, 10, 15findsg 6725 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    i^i cin 3432    C_ wss 3433   (/)c0 3758   ifcif 3906   U.cuni 4213   ran crn 4846   suc csuc 5435   ` cfv 5592    |-> cmpt2 6298   omcom 6697  seq𝜔cseqom 7163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-seqom 7164
This theorem is referenced by:  fin23lem16  8754
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