MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem12 Unicode version

Theorem ackbij1lem12 8067
Description: Lemma for ackbij1 8074. (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 8065 . . . 4  |-  F :
( ~P om  i^i  Fin ) --> om
31ackbij1lem11 8066 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  A  e.  ( ~P om  i^i  Fin ) )
4 ffvelrn 5827 . . . 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 3435 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  C_  B
)
71ackbij1lem11 8066 . . . . 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 5827 . . . 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 6831 . . 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 3660 . . . . 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 8064 . . . 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 3668 . . . . . 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 5691 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( F `
 B ) )
2116, 20eqtr3d 2438 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( ( F `
 A )  +o  ( F `  ( B  \  A ) ) )  =  ( F `
 B ) )
2212, 21sseqtrd 3344 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 1649    e. wcel 1721    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U_ciun 4053    e. cmpt 4226   omcom 4804    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040    +o coa 6680   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  ackbij1lem15  8070  ackbij1b  8075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004
  Copyright terms: Public domain W3C validator