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Theorem ackbij1lem12 8111
Description: Lemma for ackbij1 8118. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
Assertion
Ref Expression
ackbij1lem12  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  ( F `  B )
)
Distinct variable groups:    x, F, y    x, A, y    x, B, y

Proof of Theorem ackbij1lem12
StepHypRef Expression
1 ackbij.f . . . . 5  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
21ackbij1lem10 8109 . . . 4  |-  F :
( ~P om  i^i  Fin ) --> om
31ackbij1lem11 8110 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  A  e.  ( ~P om  i^i  Fin ) )
4 ffvelrn 5868 . . . 4  |-  ( ( F : ( ~P
om  i^i  Fin ) --> om  /\  A  e.  ( ~P om  i^i  Fin ) )  ->  ( F `  A )  e.  om )
52, 3, 4sylancr 645 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  e.  om )
6 difssd 3475 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  C_  B
)
71ackbij1lem11 8110 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  ( B  \  A
)  C_  B )  ->  ( B  \  A
)  e.  ( ~P
om  i^i  Fin )
)
86, 7syldan 457 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  e.  ( ~P om  i^i  Fin ) )
9 ffvelrn 5868 . . . 4  |-  ( ( F : ( ~P
om  i^i  Fin ) --> om  /\  ( B  \  A )  e.  ( ~P om  i^i  Fin ) )  ->  ( F `  ( B  \  A ) )  e. 
om )
102, 8, 9sylancr 645 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( B  \  A ) )  e.  om )
11 nnaword1 6872 . . 3  |-  ( ( ( F `  A
)  e.  om  /\  ( F `  ( B 
\  A ) )  e.  om )  -> 
( F `  A
)  C_  ( ( F `  A )  +o  ( F `  ( B  \  A ) ) ) )
125, 10, 11syl2anc 643 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  (
( F `  A
)  +o  ( F `
 ( B  \  A ) ) ) )
13 disjdif 3700 . . . . 5  |-  ( A  i^i  ( B  \  A ) )  =  (/)
1413a1i 11 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
151ackbij1lem9 8108 . . . 4  |-  ( ( A  e.  ( ~P
om  i^i  Fin )  /\  ( B  \  A
)  e.  ( ~P
om  i^i  Fin )  /\  ( A  i^i  ( B  \  A ) )  =  (/) )  ->  ( F `  ( A  u.  ( B  \  A
) ) )  =  ( ( F `  A )  +o  ( F `  ( B  \  A ) ) ) )
163, 8, 14, 15syl3anc 1184 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( ( F `  A )  +o  ( F `  ( B  \  A ) ) ) )
17 undif 3708 . . . . . 6  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  =  B )
1817biimpi 187 . . . . 5  |-  ( A 
C_  B  ->  ( A  u.  ( B  \  A ) )  =  B )
1918adantl 453 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( A  u.  ( B  \  A ) )  =  B )
2019fveq2d 5732 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( F `
 B ) )
2116, 20eqtr3d 2470 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( ( F `
 A )  +o  ( F `  ( B  \  A ) ) )  =  ( F `
 B ) )
2212, 21sseqtrd 3384 1  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  ( F `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266   omcom 4845    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    +o coa 6721   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  ackbij1lem15  8114  ackbij1b  8119
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-card 7826  df-cda 8048
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