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Theorem hashimarn 12545
Description: The size of the image of a one-to-one function  E under the range of a function  F which is a one-to-one function into the domain of  E equals the size of the function  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
Assertion
Ref Expression
hashimarn  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ( # `
 ( E " ran  F ) )  =  ( # `  F
) ) )

Proof of Theorem hashimarn
StepHypRef Expression
1 f1f 5764 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 frn 5720 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ran 
F  C_  dom  E )
31, 2syl 17 . . . . . 6  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ran  F 
C_  dom  E )
43adantl 464 . . . . 5  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  F  C_  dom  E )
5 ssdmres 5115 . . . . 5  |-  ( ran 
F  C_  dom  E  <->  dom  ( E  |`  ran  F )  =  ran  F )
64, 5sylib 196 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  dom  ( E  |`  ran  F
)  =  ran  F
)
76fveq2d 5853 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  dom  ( E  |`  ran  F ) )  =  ( # `  ran  F ) )
8 f1fun 5766 . . . . . . . . 9  |-  ( E : dom  E -1-1-> ran  E  ->  Fun  E )
9 funres 5608 . . . . . . . . . 10  |-  ( Fun 
E  ->  Fun  ( E  |`  ran  F ) )
10 funfn 5598 . . . . . . . . . 10  |-  ( Fun  ( E  |`  ran  F
)  <->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F
) )
119, 10sylib 196 . . . . . . . . 9  |-  ( Fun 
E  ->  ( E  |` 
ran  F )  Fn 
dom  ( E  |`  ran  F ) )
128, 11syl 17 . . . . . . . 8  |-  ( E : dom  E -1-1-> ran  E  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F
) )
1312adantr 463 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F ) )
1413adantr 463 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( E  |`  ran  F
)  Fn  dom  ( E  |`  ran  F ) )
15 hashfn 12491 . . . . . 6  |-  ( ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F )  -> 
( # `  ( E  |`  ran  F ) )  =  ( # `  dom  ( E  |`  ran  F
) ) )
1614, 15syl 17 . . . . 5  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E  |`  ran  F ) )  =  ( # `  dom  ( E  |`  ran  F
) ) )
17 fdm 5718 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  =  ( 0..^ ( # `  F
) ) )
18 ovex 6306 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  e.  _V
1917, 18syl6eqel 2498 . . . . . . . . 9  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  e.  _V )
201, 19syl 17 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  dom  F  e.  _V )
21 f1fun 5766 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  Fun  F )
22 funrnex 6751 . . . . . . . 8  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
2320, 21, 22sylc 59 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ran  F  e.  _V )
2423adantl 464 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  F  e.  _V )
25 rnexg 6716 . . . . . . . 8  |-  ( E  e.  V  ->  ran  E  e.  _V )
2625adantl 464 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ran  E  e.  _V )
2726adantr 463 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  E  e.  _V )
28 simpll 752 . . . . . . 7  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  E : dom  E -1-1-> ran  E )
29 f1ssres 5771 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  ran  F  C_  dom  E )  ->  ( E  |`  ran  F ) : ran  F -1-1-> ran  E )
3028, 4, 29syl2anc 659 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( E  |`  ran  F
) : ran  F -1-1-> ran 
E )
31 hashf1rn 12472 . . . . . . 7  |-  ( ( ran  F  e.  _V  /\ 
ran  E  e.  _V )  ->  ( ( E  |`  ran  F ) : ran  F -1-1-> ran  E  ->  ( # `  ( E  |`  ran  F ) )  =  ( # `  ran  ( E  |`  ran  F ) ) ) )
3231imp 427 . . . . . 6  |-  ( ( ( ran  F  e. 
_V  /\  ran  E  e. 
_V )  /\  ( E  |`  ran  F ) : ran  F -1-1-> ran  E )  ->  ( # `  ( E  |`  ran  F ) )  =  ( # `  ran  ( E  |`  ran  F ) ) )
3324, 27, 30, 32syl21anc 1229 . . . . 5  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E  |`  ran  F ) )  =  ( # `  ran  ( E  |`  ran  F
) ) )
3416, 33eqtr3d 2445 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  dom  ( E  |`  ran  F ) )  =  ( # `  ran  ( E  |`  ran  F ) ) )
35 df-ima 4836 . . . . 5  |-  ( E
" ran  F )  =  ran  ( E  |`  ran  F )
3635fveq2i 5852 . . . 4  |-  ( # `  ( E " ran  F ) )  =  (
# `  ran  ( E  |`  ran  F ) )
3734, 36syl6reqr 2462 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E
" ran  F )
)  =  ( # `  dom  ( E  |`  ran  F ) ) )
3818a1i 11 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( 0..^ ( # `  F ) )  e. 
_V )
39 dmexg 6715 . . . . . 6  |-  ( E  e.  V  ->  dom  E  e.  _V )
4039adantl 464 . . . . 5  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  dom  E  e.  _V )
4140adantr 463 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  dom  E  e.  _V )
42 simpr 459 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )
43 hashf1rn 12472 . . . . 5  |-  ( ( ( 0..^ ( # `  F ) )  e. 
_V  /\  dom  E  e. 
_V )  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ( # `
 F )  =  ( # `  ran  F ) ) )
4443imp 427 . . . 4  |-  ( ( ( ( 0..^ (
# `  F )
)  e.  _V  /\  dom  E  e.  _V )  /\  F : ( 0..^ ( # `  F
) ) -1-1-> dom  E
)  ->  ( # `  F
)  =  ( # `  ran  F ) )
4538, 41, 42, 44syl21anc 1229 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  F )  =  ( # `  ran  F ) )
467, 37, 453eqtr4d 2453 . 2  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E
" ran  F )
)  =  ( # `  F ) )
4746ex 432 1  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ( # `
 ( E " ran  F ) )  =  ( # `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   dom cdm 4823   ran crn 4824    |` cres 4825   "cima 4826   Fun wfun 5563    Fn wfn 5564   -->wf 5565   -1-1->wf1 5566   ` cfv 5569  (class class class)co 6278   0cc0 9522  ..^cfzo 11854   #chash 12452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-hash 12453
This theorem is referenced by:  hashimarni  12546
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