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Theorem hashimarn 27994
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 5598 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 frn 5556 . . . . . . 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 453 . . . . 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 5127 . . . . 5  |-  ( ran 
F  C_  dom  E  <->  dom  ( E  |`  ran  F )  =  ran  F )
64, 5sylib 189 . . . 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 5691 . . 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 5600 . . . . . . . . 9  |-  ( E : dom  E -1-1-> ran  E  ->  Fun  E )
9 funres 5451 . . . . . . . . . 10  |-  ( Fun 
E  ->  Fun  ( E  |`  ran  F ) )
10 funfn 5441 . . . . . . . . . 10  |-  ( Fun  ( E  |`  ran  F
)  <->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F
) )
119, 10sylib 189 . . . . . . . . 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 452 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F ) )
1413adantr 452 . . . . . 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 11604 . . . . . 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 5554 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  =  ( 0..^ ( # `  F
) ) )
18 ovex 6065 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  e.  _V
1917, 18syl6eqel 2492 . . . . . . . . 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 5600 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  Fun  F )
22 funrnex 5926 . . . . . . . 8  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
2320, 21, 22sylc 58 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ran  F  e.  _V )
2423adantl 453 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  F  e.  _V )
25 rnexg 5090 . . . . . . . 8  |-  ( E  e.  V  ->  ran  E  e.  _V )
2625adantl 453 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ran  E  e.  _V )
2726adantr 452 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  E  e.  _V )
28 simpll 731 . . . . . . 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 5605 . . . . . . 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 643 . . . . . 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 11591 . . . . . . 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 419 . . . . . 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 1183 . . . . 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 2438 . . . 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 4850 . . . . 5  |-  ( E
" ran  F )  =  ran  ( E  |`  ran  F )
3635fveq2i 5690 . . . 4  |-  ( # `  ( E " ran  F ) )  =  (
# `  ran  ( E  |`  ran  F ) )
3734, 36syl6reqr 2455 . . 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 5089 . . . . . 6  |-  ( E  e.  V  ->  dom  E  e.  _V )
4039adantl 453 . . . . 5  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  dom  E  e.  _V )
4140adantr 452 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  dom  E  e.  _V )
42 simpr 448 . . . 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 11591 . . . . 5  |-  ( ( ( 0..^ ( # `  F ) )  e. 
_V  /\  dom  E  e. 
_V )  ->  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  ( # `
 F )  =  ( # `  ran  F ) ) )
4443imp 419 . . . 4  |-  ( ( ( ( 0..^ (
# `  F )
)  e.  _V  /\  dom  E  e.  _V )  /\  F : ( 0..^ ( # `  F
) ) -1-1-> dom  E
)  ->  ( # `  F
)  =  ( # `  ran  F ) )
4538, 41, 42, 44syl21anc 1183 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  F )  =  ( # `  ran  F ) )
467, 37, 453eqtr4d 2446 . 2  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  ( E
" ran  F )
)  =  ( # `  F ) )
4746ex 424 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   ` cfv 5413  (class class class)co 6040   0cc0 8946  ..^cfzo 11090   #chash 11573
This theorem is referenced by:  hashimarni  27995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-hash 11574
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