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Theorem hashimarn 12449
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 5772 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 frn 5728 . . . . . . 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 466 . . . . 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 5286 . . . . 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 5861 . . 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 5774 . . . . . . . . 9  |-  ( E : dom  E -1-1-> ran  E  ->  Fun  E )
9 funres 5618 . . . . . . . . . 10  |-  ( Fun 
E  ->  Fun  ( E  |`  ran  F ) )
10 funfn 5608 . . . . . . . . . 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 465 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ( E  |`  ran  F )  Fn  dom  ( E  |`  ran  F ) )
1413adantr 465 . . . . . 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 12398 . . . . . 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 5726 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  =  ( 0..^ ( # `  F
) ) )
18 ovex 6300 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  e.  _V
1917, 18syl6eqel 2556 . . . . . . . . 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 5774 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  Fun  F )
22 funrnex 6741 . . . . . . . 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 466 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  F  e.  _V )
25 rnexg 6706 . . . . . . . 8  |-  ( E  e.  V  ->  ran  E  e.  _V )
2625adantl 466 . . . . . . 7  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  ran  E  e.  _V )
2726adantr 465 . . . . . 6  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  ran  E  e.  _V )
28 simpll 753 . . . . . . 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 5779 . . . . . . 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 661 . . . . . 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 12380 . . . . . . 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 1222 . . . . 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 2503 . . . 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 5005 . . . . 5  |-  ( E
" ran  F )  =  ran  ( E  |`  ran  F )
3635fveq2i 5860 . . . 4  |-  ( # `  ( E " ran  F ) )  =  (
# `  ran  ( E  |`  ran  F ) )
3734, 36syl6reqr 2520 . . 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 6705 . . . . . 6  |-  ( E  e.  V  ->  dom  E  e.  _V )
4039adantl 466 . . . . 5  |-  ( ( E : dom  E -1-1-> ran 
E  /\  E  e.  V )  ->  dom  E  e.  _V )
4140adantr 465 . . . 4  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  ->  dom  E  e.  _V )
42 simpr 461 . . . 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 12380 . . . . 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 1222 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  F )  =  ( # `  ran  F ) )
467, 37, 453eqtr4d 2511 . 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 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573    Fn wfn 5574   -->wf 5575   -1-1->wf1 5576   ` cfv 5579  (class class class)co 6275   0cc0 9481  ..^cfzo 11781   #chash 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-hash 12361
This theorem is referenced by:  hashimarni  12450
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