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Theorem ackbij1lem12 7873
Description: Lemma for ackbij1 7880. (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 7871 . . . 4  |-  F :
( ~P om  i^i  Fin ) --> om
31ackbij1lem11 7872 . . . 4  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  A  e.  ( ~P om  i^i  Fin ) )
4 ffvelrn 5679 . . . 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 3316 . . . . . 6  |-  ( B 
\  A )  C_  B
76a1i 10 . . . . 5  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( B  \  A )  C_  B
)
81ackbij1lem11 7872 . . . . 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 5679 . . . 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 6643 . . 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 3539 . . . . 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 7870 . . . 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 3547 . . . . . 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 5545 . . 3  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( F `  ( A  u.  ( B  \  A ) ) )  =  ( F `
 B ) )
2217, 21eqtr3d 2330 . 2  |-  ( ( B  e.  ( ~P
om  i^i  Fin )  /\  A  C_  B )  ->  ( ( F `
 A )  +o  ( F `  ( B  \  A ) ) )  =  ( F `
 B ) )
2313, 22sseqtrd 3227 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 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U_ciun 3921    e. cmpt 4093   omcom 4672    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    +o coa 6492   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  ackbij1lem15  7876  ackbij1b  7881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810
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