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Theorem hashimarn 12184
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 5594 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 frn 5553 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ran 
F  C_  dom  E )
31, 2syl 16 . . . . . 6  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ran  F 
C_  dom  E )
43adantl 463 . . . . 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 5120 . . . . 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 5683 . . 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 5596 . . . . . . . . 9  |-  ( E : dom  E -1-1-> ran  E  ->  Fun  E )
9 funres 5445 . . . . . . . . . 10  |-  ( Fun 
E  ->  Fun  ( E  |`  ran  F ) )
10 funfn 5435 . . . . . . . . . 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 16 . . . . . . . 8  |-  ( E : dom  E -1-1-> ran  E  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F
) )
1312adantr 462 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F ) )
1413adantr 462 . . . . . 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 12122 . . . . . 6  |-  ( ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F )  -> 
( # `  ( E  |`  ran  F ) )  =  ( # `  dom  ( E  |`  ran  F
) ) )
1614, 15syl 16 . . . . 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 5551 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  =  ( 0..^ ( # `  F
) ) )
18 ovex 6105 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  e.  _V
1917, 18syl6eqel 2521 . . . . . . . . 9  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  e.  _V )
201, 19syl 16 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  dom  F  e.  _V )
21 f1fun 5596 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  Fun  F )
22 funrnex 6533 . . . . . . . 8  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
2320, 21, 22sylc 60 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ran  F  e.  _V )
2423adantl 463 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  F  e.  _V )
25 rnexg 6499 . . . . . . . 8  |-  ( E  e.  V  ->  ran  E  e.  _V )
2625adantl 463 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ran  E  e.  _V )
2726adantr 462 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  E  e.  _V )
28 simpll 746 . . . . . . 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 5601 . . . . . . 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 654 . . . . . 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 12107 . . . . . . 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 429 . . . . . 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 1210 . . . . 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 2467 . . . 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 4840 . . . . 5  |-  ( E
" ran  F )  =  ran  ( E  |`  ran  F )
3635fveq2i 5682 . . . 4  |-  ( # `  ( E " ran  F ) )  =  (
# `  ran  ( E  |`  ran  F ) )
3734, 36syl6reqr 2484 . . 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 6498 . . . . . 6  |-  ( E  e.  V  ->  dom  E  e.  _V )
4039adantl 463 . . . . 5  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  dom  E  e.  _V )
4140adantr 462 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  dom  E  e.  _V )
42 simpr 458 . . . 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 12107 . . . . 5  |-  ( ( ( 0..^ ( # `  F ) )  e. 
_V  /\  dom  E  e. 
_V )  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ( # `
 F )  =  ( # `  ran  F ) ) )
4443imp 429 . . . 4  |-  ( ( ( ( 0..^ (
# `  F )
)  e.  _V  /\  dom  E  e.  _V )  /\  F : ( 0..^ ( # `  F
) ) -1-1-> dom  E
)  ->  ( # `  F
)  =  ( # `  ran  F ) )
4538, 41, 42, 44syl21anc 1210 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  F )  =  ( # `  ran  F ) )
467, 37, 453eqtr4d 2475 . 2  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E
" ran  F )
)  =  ( # `  F ) )
4746ex 434 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 369    = wceq 1362    e. wcel 1755   _Vcvv 2962    C_ wss 3316   dom cdm 4827   ran crn 4828    |` cres 4829   "cima 4830   Fun wfun 5400    Fn wfn 5401   -->wf 5402   -1-1->wf1 5403   ` cfv 5406  (class class class)co 6080   0cc0 9270  ..^cfzo 11532   #chash 12087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-hash 12088
This theorem is referenced by:  hashimarni  12185
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