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Theorem hashun3 11366
Description: The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.)
Assertion
Ref Expression
hashun3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( ( ( # `  A
)  +  ( # `  B ) )  -  ( # `  ( A  i^i  B ) ) ) )

Proof of Theorem hashun3
StepHypRef Expression
1 diffi 7089 . . . . . . 7  |-  ( B  e.  Fin  ->  ( B  \  A )  e. 
Fin )
21adantl 452 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( B  \  A
)  e.  Fin )
3 simpl 443 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  A  e.  Fin )
4 inss1 3389 . . . . . . 7  |-  ( A  i^i  B )  C_  A
5 ssfi 7083 . . . . . . 7  |-  ( ( A  e.  Fin  /\  ( A  i^i  B ) 
C_  A )  -> 
( A  i^i  B
)  e.  Fin )
63, 4, 5sylancl 643 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  B
)  e.  Fin )
7 sslin 3395 . . . . . . . . 9  |-  ( ( A  i^i  B ) 
C_  A  ->  (
( B  \  A
)  i^i  ( A  i^i  B ) )  C_  ( ( B  \  A )  i^i  A
) )
84, 7ax-mp 8 . . . . . . . 8  |-  ( ( B  \  A )  i^i  ( A  i^i  B ) )  C_  (
( B  \  A
)  i^i  A )
9 incom 3361 . . . . . . . . 9  |-  ( ( B  \  A )  i^i  A )  =  ( A  i^i  ( B  \  A ) )
10 disjdif 3526 . . . . . . . . 9  |-  ( A  i^i  ( B  \  A ) )  =  (/)
119, 10eqtri 2303 . . . . . . . 8  |-  ( ( B  \  A )  i^i  A )  =  (/)
12 sseq0 3486 . . . . . . . 8  |-  ( ( ( ( B  \  A )  i^i  ( A  i^i  B ) ) 
C_  ( ( B 
\  A )  i^i 
A )  /\  (
( B  \  A
)  i^i  A )  =  (/) )  ->  (
( B  \  A
)  i^i  ( A  i^i  B ) )  =  (/) )
138, 11, 12mp2an 653 . . . . . . 7  |-  ( ( B  \  A )  i^i  ( A  i^i  B ) )  =  (/)
1413a1i 10 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  \  A )  i^i  ( A  i^i  B ) )  =  (/) )
15 hashun 11364 . . . . . 6  |-  ( ( ( B  \  A
)  e.  Fin  /\  ( A  i^i  B )  e.  Fin  /\  (
( B  \  A
)  i^i  ( A  i^i  B ) )  =  (/) )  ->  ( # `  ( ( B  \  A )  u.  ( A  i^i  B ) ) )  =  ( (
# `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) ) )
162, 6, 14, 15syl3anc 1182 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( (
# `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) ) )
17 incom 3361 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
1817uneq2i 3326 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  ( ( B  \  A
)  u.  ( B  i^i  A ) )
19 uncom 3319 . . . . . . . 8  |-  ( ( B  \  A )  u.  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  u.  ( B 
\  A ) )
20 inundif 3532 . . . . . . . 8  |-  ( ( B  i^i  A )  u.  ( B  \  A ) )  =  B
2118, 19, 203eqtri 2307 . . . . . . 7  |-  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  B
2221a1i 10 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( B  \  A )  u.  ( A  i^i  B ) )  =  B )
2322fveq2d 5529 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  (
( B  \  A
)  u.  ( A  i^i  B ) ) )  =  ( # `  B ) )
2416, 23eqtr3d 2317 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) )  =  ( # `  B ) )
25 hashcl 11350 . . . . . . 7  |-  ( B  e.  Fin  ->  ( # `
 B )  e. 
NN0 )
2625adantl 452 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  NN0 )
2726nn0cnd 10020 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  B
)  e.  CC )
28 hashcl 11350 . . . . . . 7  |-  ( ( A  i^i  B )  e.  Fin  ->  ( # `
 ( A  i^i  B ) )  e.  NN0 )
296, 28syl 15 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  i^i  B ) )  e.  NN0 )
3029nn0cnd 10020 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  i^i  B ) )  e.  CC )
31 hashcl 11350 . . . . . . 7  |-  ( ( B  \  A )  e.  Fin  ->  ( # `
 ( B  \  A ) )  e. 
NN0 )
322, 31syl 15 . . . . . 6  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  NN0 )
3332nn0cnd 10020 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( B  \  A ) )  e.  CC )
3427, 30, 33subadd2d 9176 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( # `  B )  -  ( # `
 ( A  i^i  B ) ) )  =  ( # `  ( B  \  A ) )  <-> 
( ( # `  ( B  \  A ) )  +  ( # `  ( A  i^i  B ) ) )  =  ( # `  B ) ) )
3524, 34mpbird 223 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  B
)  -  ( # `  ( A  i^i  B
) ) )  =  ( # `  ( B  \  A ) ) )
3635oveq2d 5874 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) )  =  ( ( # `  A
)  +  ( # `  ( B  \  A
) ) ) )
37 hashcl 11350 . . . . 5  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
3837adantr 451 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  NN0 )
3938nn0cnd 10020 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  A
)  e.  CC )
4039, 27, 30addsubassd 9177 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( # `  A )  +  (
# `  B )
)  -  ( # `  ( A  i^i  B
) ) )  =  ( ( # `  A
)  +  ( (
# `  B )  -  ( # `  ( A  i^i  B ) ) ) ) )
41 undif2 3530 . . . 4  |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B
)
4241fveq2i 5528 . . 3  |-  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( # `  ( A  u.  B
) )
4310a1i 10 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
44 hashun 11364 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  \  A )  e.  Fin  /\  ( A  i^i  ( B  \  A ) )  =  (/) )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
453, 2, 43, 44syl3anc 1182 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  ( B  \  A ) ) )  =  ( ( # `  A )  +  (
# `  ( B  \  A ) ) ) )
4642, 45syl5eqr 2329 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( (
# `  A )  +  ( # `  ( B  \  A ) ) ) )
4736, 40, 463eqtr4rd 2326 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B )
)  =  ( ( ( # `  A
)  +  ( # `  B ) )  -  ( # `  ( A  i^i  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   ` cfv 5255  (class class class)co 5858   Fincfn 6863    + caddc 8740    - cmin 9037   NN0cn0 9965   #chash 11337
This theorem is referenced by:  incexclem  12295
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-hash 11338
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