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

Theorem ackbij1lem12 7857
Description: Lemma for ackbij1 7864. (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 7855 . . . 4  |-  F :
( ~P om  i^i  Fin ) --> om
31ackbij1lem11 7856 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  A  e.  ( ~P om  i^i  Fin ) )
4 ffvelrn 5663 . . . 4  |-  ( ( F : ( ~P
om  i^i  Fin ) --> om  /\  A  e.  ( ~P om  i^i  Fin ) )  ->  ( F `  A )  e.  om )
52, 3, 4sylancr 644 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  e.  om )
6 difss 3303 . . . . . 6  |-  ( B 
\  A )  C_  B
76a1i 10 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  C_  B
)
81ackbij1lem11 7856 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  ( B  \  A
)  C_  B )  ->  ( B  \  A
)  e.  ( ~P
om  i^i  Fin )
)
97, 8syldan 456 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  e.  ( ~P om  i^i  Fin ) )
10 ffvelrn 5663 . . . 4  |-  ( ( F : ( ~P
om  i^i  Fin ) --> om  /\  ( B  \  A )  e.  ( ~P om  i^i  Fin ) )  ->  ( F `  ( B  \  A ) )  e. 
om )
112, 9, 10sylancr 644 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( B  \  A ) )  e.  om )
12 nnaword1 6627 . . 3  |-  ( ( ( F `  A
)  e.  om  /\  ( F `  ( B 
\  A ) )  e.  om )  -> 
( F `  A
)  C_  ( ( F `  A )  +o  ( F `  ( B  \  A ) ) ) )
135, 11, 12syl2anc 642 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  A )  C_  (
( F `  A
)  +o  ( F `
 ( B  \  A ) ) ) )
14 disjdif 3526 . . . . 5  |-  ( A  i^i  ( B  \  A ) )  =  (/)
1514a1i 10 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
161ackbij1lem9 7854 . . . 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 ) ) ) )
173, 9, 15, 16syl3anc 1182 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( ( F `  A )  +o  ( F `  ( B  \  A ) ) ) )
18 undif 3534 . . . . . 6  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  =  B )
1918biimpi 186 . . . . 5  |-  ( A 
C_  B  ->  ( A  u.  ( B  \  A ) )  =  B )
2019adantl 452 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( A  u.  ( B  \  A ) )  =  B )
2120fveq2d 5529 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( F `
 B ) )
2217, 21eqtr3d 2317 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( ( F `
 A )  +o  ( F `  ( B  \  A ) ) )  =  ( F `
 B ) )
2313, 22sseqtrd 3214 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 358    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U_ciun 3905    e. cmpt 4077   omcom 4656    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    +o coa 6476   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ackbij1lem15  7860  ackbij1b  7865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794
  Copyright terms: Public domain W3C validator