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Theorem hashimarn 12302
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 5704 . . . . . . 7  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 frn 5663 . . . . . . 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 5230 . . . . 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 5793 . . 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 5706 . . . . . . . . 9  |-  ( E : dom  E -1-1-> ran  E  ->  Fun  E )
9 funres 5555 . . . . . . . . . 10  |-  ( Fun 
E  ->  Fun  ( E  |`  ran  F ) )
10 funfn 5545 . . . . . . . . . 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 12240 . . . . . 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 5661 . . . . . . . . . 10  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  dom 
F  =  ( 0..^ ( # `  F
) ) )
18 ovex 6215 . . . . . . . . . 10  |-  ( 0..^ ( # `  F
) )  e.  _V
1917, 18syl6eqel 2547 . . . . . . . . 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 5706 . . . . . . . 8  |-  ( F : ( 0..^ (
# `  F )
) -1-1-> dom  E  ->  Fun  F )
22 funrnex 6644 . . . . . . . 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 6610 . . . . . . . 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 5711 . . . . . . 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 12224 . . . . . . 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 1218 . . . . 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 2494 . . . 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 4951 . . . . 5  |-  ( E
" ran  F )  =  ran  ( E  |`  ran  F )
3635fveq2i 5792 . . . 4  |-  ( # `  ( E " ran  F ) )  =  (
# `  ran  ( E  |`  ran  F ) )
3734, 36syl6reqr 2511 . . 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 6609 . . . . . 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 12224 . . . . 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 1218 . . 3  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  E  e.  V )  /\  F : ( 0..^ (
# `  F )
) -1-1-> dom  E )  -> 
( # `  F )  =  ( # `  ran  F ) )
467, 37, 453eqtr4d 2502 . 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 1370    e. wcel 1758   _Vcvv 3068    C_ wss 3426   dom cdm 4938   ran crn 4939    |` cres 4940   "cima 4941   Fun wfun 5510    Fn wfn 5511   -->wf 5512   -1-1->wf1 5513   ` cfv 5516  (class class class)co 6190   0cc0 9383  ..^cfzo 11649   #chash 12204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-hash 12205
This theorem is referenced by:  hashimarni  12303
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