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Theorem ackbij2lem4 8122
Description: Lemma for ackbij2 8123. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypotheses
Ref Expression
ackbij.f  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
ackbij.g  |-  G  =  ( x  e.  _V  |->  ( y  e.  ~P dom  x  |->  ( F `  ( x " y
) ) ) )
Assertion
Ref Expression
ackbij2lem4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  A )
)
Distinct variable groups:    x, F, y    x, G, y    x, A, y    x, B, y

Proof of Theorem ackbij2lem4
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . 3  |-  ( a  =  B  ->  ( rec ( G ,  (/) ) `  a )  =  ( rec ( G ,  (/) ) `  B ) )
21sseq2d 3376 . 2  |-  ( a  =  B  ->  (
( rec ( G ,  (/) ) `  B
)  C_  ( rec ( G ,  (/) ) `  a )  <->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  B )
) )
3 fveq2 5728 . . 3  |-  ( a  =  b  ->  ( rec ( G ,  (/) ) `  a )  =  ( rec ( G ,  (/) ) `  b ) )
43sseq2d 3376 . 2  |-  ( a  =  b  ->  (
( rec ( G ,  (/) ) `  B
)  C_  ( rec ( G ,  (/) ) `  a )  <->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  b )
) )
5 fveq2 5728 . . 3  |-  ( a  =  suc  b  -> 
( rec ( G ,  (/) ) `  a
)  =  ( rec ( G ,  (/) ) `  suc  b ) )
65sseq2d 3376 . 2  |-  ( a  =  suc  b  -> 
( ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  a )  <->  ( rec ( G ,  (/) ) `  B ) 
C_  ( rec ( G ,  (/) ) `  suc  b ) ) )
7 fveq2 5728 . . 3  |-  ( a  =  A  ->  ( rec ( G ,  (/) ) `  a )  =  ( rec ( G ,  (/) ) `  A ) )
87sseq2d 3376 . 2  |-  ( a  =  A  ->  (
( rec ( G ,  (/) ) `  B
)  C_  ( rec ( G ,  (/) ) `  a )  <->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  A )
) )
9 ssid 3367 . . 3  |-  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  B
)
109a1i 11 . 2  |-  ( B  e.  om  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  B
) )
11 ackbij.f . . . . 5  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
12 ackbij.g . . . . 5  |-  G  =  ( x  e.  _V  |->  ( y  e.  ~P dom  x  |->  ( F `  ( x " y
) ) ) )
1311, 12ackbij2lem3 8121 . . . 4  |-  ( b  e.  om  ->  ( rec ( G ,  (/) ) `  b )  C_  ( rec ( G ,  (/) ) `  suc  b ) )
1413ad2antrr 707 . . 3  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( rec ( G ,  (/) ) `  b )  C_  ( rec ( G ,  (/) ) `  suc  b ) )
15 sstr2 3355 . . 3  |-  ( ( rec ( G ,  (/) ) `  B ) 
C_  ( rec ( G ,  (/) ) `  b )  ->  (
( rec ( G ,  (/) ) `  b
)  C_  ( rec ( G ,  (/) ) `  suc  b )  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  suc  b ) ) )
1614, 15syl5com 28 . 2  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  b
)  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  suc  b ) ) )
172, 4, 6, 8, 10, 16findsg 4872 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( rec ( G ,  (/) ) `  B )  C_  ( rec ( G ,  (/) ) `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266   suc csuc 4583   omcom 4845    X. cxp 4876   dom cdm 4878   "cima 4881   ` cfv 5454   reccrdg 6667   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  ackbij2  8123
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-r1 7690  df-card 7826  df-cda 8048
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