Theorem hashimarn 13227

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.) (Proof shortened by AV, 4-May-2021.)

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

Step	Hyp	Ref	Expression
1		f1f 6101	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
2		frn 6053	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) --> dom E -> ran F C_ dom E )
3	1, 2	syl 17	. . . . . 6 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> ran F C_ dom E )
4	3	adantl 482	. . . . 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 5420	. . . . 5 |- ( ran F C_ dom E <-> dom ( E |` ran F ) = ran F )
6	4, 5	sylib 208	. . . 4 |- ( ( ( E : dom E -1-1-> ran E /\ E e. V ) /\ F : ( 0..^ ( # ` F ) ) -1-1-> dom E ) -> dom ( E |` ran F ) = ran F )
7	6	fveq2d 6195	. . 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 6103	. . . . . . . 8 |- ( E : dom E -1-1-> ran E -> Fun E )
9		funres 5929	. . . . . . . . 9 |- ( Fun E -> Fun ( E |` ran F ) )
10		funfn 5918	. . . . . . . . 9 |- ( Fun ( E |` ran F ) <-> ( E |` ran F ) Fn dom ( E |` ran F ) )
11	9, 10	sylib 208	. . . . . . . 8 |- ( Fun E -> ( E |` ran F ) Fn dom ( E |` ran F ) )
12	8, 11	syl 17	. . . . . . 7 |- ( E : dom E -1-1-> ran E -> ( E |` ran F ) Fn dom ( E |` ran F ) )
13	12	ad2antrr 762	. . . . . 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 ) )
14		hashfn 13164	. . . . . 6 |- ( ( E |` ran F ) Fn dom ( E |` ran F ) -> ( # ` ( E |` ran F ) ) = ( # ` dom ( E |` ran F ) ) )
15	13, 14	syl 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 ) ) )
16		ovex 6678	. . . . . . . 8 |- ( 0..^ ( # ` F ) ) e. _V
17		fex 6490	. . . . . . . 8 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( 0..^ ( # ` F ) ) e. _V ) -> F e. _V )
18	1, 16, 17	sylancl 694	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> F e. _V )
19		rnexg 7098	. . . . . . 7 |- ( F e. _V -> ran F e. _V )
20	18, 19	syl 17	. . . . . 6 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> ran F e. _V )
21		simpll 790	. . . . . . 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 )
22		f1ssres 6108	. . . . . . 7 |- ( ( E : dom E -1-1-> ran E /\ ran F C_ dom E ) -> ( E |` ran F ) : ran F -1-1-> ran E )
23	21, 4, 22	syl2anc 693	. . . . . 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 )
24		hashf1rn 13142	. . . . . 6 |- ( ( ran F e. _V /\ ( E |` ran F ) : ran F -1-1-> ran E ) -> ( # ` ( E |` ran F ) ) = ( # ` ran ( E |` ran F ) ) )
25	20, 23, 24	syl2an2 875	. . . . 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 ) ) )
26	15, 25	eqtr3d 2658	. . . 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 ) ) )
27		df-ima 5127	. . . . 5 |- ( E " ran F ) = ran ( E |` ran F )
28	27	fveq2i 6194	. . . 4 |- ( # ` ( E " ran F ) ) = ( # ` ran ( E |` ran F ) )
29	26, 28	syl6reqr 2675	. . 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 ) ) )
30		hashf1rn 13142	. . . . 5 |- ( ( ( 0..^ ( # ` F ) ) e. _V /\ F : ( 0..^ ( # ` F ) ) -1-1-> dom E ) -> ( # ` F ) = ( # ` ran F ) )
31	16, 30	mpan 706	. . . 4 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> ( # ` F ) = ( # ` ran F ) )
32	31	adantl 482	. . 3 |- ( ( ( E : dom E -1-1-> ran E /\ E e. V ) /\ F : ( 0..^ ( # ` F ) ) -1-1-> dom E ) -> ( # ` F ) = ( # ` ran F ) )
33	7, 29, 32	3eqtr4d 2666	. 2 |- ( ( ( E : dom E -1-1-> ran E /\ E e. V ) /\ F : ( 0..^ ( # ` F ) ) -1-1-> dom E ) -> ( # ` ( E " ran F ) ) = ( # ` F ) )
34	33	ex 450	1 |- ( ( E : dom E -1-1-> ran E /\ E e. V ) -> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E -> ( # ` ( E " ran F ) ) = ( # ` F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-hash 13118

This theorem is referenced by:  hashimarni  13228