Theorem fmfnfmlem3 21020

Description: Lemma for fmfnfm 21022. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypotheses

Ref	Expression
fmfnfm.b	|- ( ph -> B e. ( fBas ` Y ) )
fmfnfm.l	|- ( ph -> L e. ( Fil ` X ) )
fmfnfm.f	|- ( ph -> F : Y --> X )
fmfnfm.fm	|- ( ph -> ( ( X FilMap F ) ` B ) C_ L )

Assertion

Ref	Expression
fmfnfmlem3	|- ( ph -> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) )

Distinct variable groups:   x, B   x, F   x, L   ph, x   x, X   x, Y

Proof of Theorem fmfnfmlem3

Dummy variables s t y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmfnfm.l	. . . . . . . 8 |- ( ph -> L e. ( Fil ` X ) )
2		filin 20918	. . . . . . . . 9 |- ( ( L e. ( Fil ` X ) /\ y e. L /\ z e. L ) -> ( y i^i z ) e. L )
3	2	3expb 1216	. . . . . . . 8 |- ( ( L e. ( Fil ` X ) /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
4	1, 3	sylan 478	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
5		fmfnfm.f	. . . . . . . . 9 |- ( ph -> F : Y --> X )
6		ffun 5754	. . . . . . . . 9 |- ( F : Y --> X -> Fun F )
7		funcnvcnv 5663	. . . . . . . . 9 |- ( Fun F -> Fun `' `' F )
8		imain 5681	. . . . . . . . . 10 |- ( Fun `' `' F -> ( `' F " ( y i^i z ) ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
9	8	eqcomd 2468	. . . . . . . . 9 |- ( Fun `' `' F -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
10	5, 6, 7, 9	4syl 19	. . . . . . . 8 |- ( ph -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
11	10	adantr 471	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
12		imaeq2 5183	. . . . . . . . 9 |- ( x = ( y i^i z ) -> ( `' F " x ) = ( `' F " ( y i^i z ) ) )
13	12	eqeq2d 2472	. . . . . . . 8 |- ( x = ( y i^i z ) -> ( ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) ) )
14	13	rspcev 3162	. . . . . . 7 |- ( ( ( y i^i z ) e. L /\ ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) ) -> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) )
15	4, 11, 14	syl2anc 671	. . . . . 6 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) )
16		ineq12 3641	. . . . . . . 8 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( s i^i t ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
17	16	eqeq1d 2464	. . . . . . 7 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( ( s i^i t ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
18	17	rexbidv 2913	. . . . . 6 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( E. x e. L ( s i^i t ) = ( `' F " x ) <-> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
19	15, 18	syl5ibrcom 230	. . . . 5 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> E. x e. L ( s i^i t ) = ( `' F " x ) ) )
20	19	rexlimdvva 2898	. . . 4 |- ( ph -> ( E. y e. L E. z e. L ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> E. x e. L ( s i^i t ) = ( `' F " x ) ) )
21		imaeq2 5183	. . . . . . . 8 |- ( x = y -> ( `' F " x ) = ( `' F " y ) )
22	21	eqeq2d 2472	. . . . . . 7 |- ( x = y -> ( s = ( `' F " x ) <-> s = ( `' F " y ) ) )
23	22	cbvrexv 3032	. . . . . 6 |- ( E. x e. L s = ( `' F " x ) <-> E. y e. L s = ( `' F " y ) )
24		imaeq2 5183	. . . . . . . 8 |- ( x = z -> ( `' F " x ) = ( `' F " z ) )
25	24	eqeq2d 2472	. . . . . . 7 |- ( x = z -> ( t = ( `' F " x ) <-> t = ( `' F " z ) ) )
26	25	cbvrexv 3032	. . . . . 6 |- ( E. x e. L t = ( `' F " x ) <-> E. z e. L t = ( `' F " z ) )
27	23, 26	anbi12i 708	. . . . 5 |- ( ( E. x e. L s = ( `' F " x ) /\ E. x e. L t = ( `' F " x ) ) <-> ( E. y e. L s = ( `' F " y ) /\ E. z e. L t = ( `' F " z ) ) )
28		vex 3060	. . . . . . 7 |- s e. _V
29		eqid 2462	. . . . . . . 8 |- ( x e. L |-> ( `' F " x ) ) = ( x e. L |-> ( `' F " x ) )
30	29	elrnmpt 5100	. . . . . . 7 |- ( s e. _V -> ( s e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L s = ( `' F " x ) ) )
31	28, 30	ax-mp 5	. . . . . 6 |- ( s e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L s = ( `' F " x ) )
32		vex 3060	. . . . . . 7 |- t e. _V
33	29	elrnmpt 5100	. . . . . . 7 |- ( t e. _V -> ( t e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L t = ( `' F " x ) ) )
34	32, 33	ax-mp 5	. . . . . 6 |- ( t e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L t = ( `' F " x ) )
35	31, 34	anbi12i 708	. . . . 5 |- ( ( s e. ran ( x e. L |-> ( `' F " x ) ) /\ t e. ran ( x e. L |-> ( `' F " x ) ) ) <-> ( E. x e. L s = ( `' F " x ) /\ E. x e. L t = ( `' F " x ) ) )
36		reeanv 2970	. . . . 5 |- ( E. y e. L E. z e. L ( s = ( `' F " y ) /\ t = ( `' F " z ) ) <-> ( E. y e. L s = ( `' F " y ) /\ E. z e. L t = ( `' F " z ) ) )
37	27, 35, 36	3bitr4i 285	. . . 4 |- ( ( s e. ran ( x e. L |-> ( `' F " x ) ) /\ t e. ran ( x e. L |-> ( `' F " x ) ) ) <-> E. y e. L E. z e. L ( s = ( `' F " y ) /\ t = ( `' F " z ) ) )
38	28	inex1 4558	. . . . 5 |- ( s i^i t ) e. _V
39	29	elrnmpt 5100	. . . . 5 |- ( ( s i^i t ) e. _V -> ( ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L ( s i^i t ) = ( `' F " x ) ) )
40	38, 39	ax-mp 5	. . . 4 |- ( ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) <-> E. x e. L ( s i^i t ) = ( `' F " x ) )
41	20, 37, 40	3imtr4g 278	. . 3 |- ( ph -> ( ( s e. ran ( x e. L |-> ( `' F " x ) ) /\ t e. ran ( x e. L |-> ( `' F " x ) ) ) -> ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) ) )
42	41	ralrimivv 2820	. 2 |- ( ph -> A. s e. ran ( x e. L |-> ( `' F " x ) ) A. t e. ran ( x e. L |-> ( `' F " x ) ) ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) )
43		mptexg 6160	. . 3 |- ( L e. ( Fil ` X ) -> ( x e. L |-> ( `' F " x ) ) e. _V )
44		rnexg 6752	. . 3 |- ( ( x e. L |-> ( `' F " x ) ) e. _V -> ran ( x e. L |-> ( `' F " x ) ) e. _V )
45		inficl 7965	. . 3 |- ( ran ( x e. L |-> ( `' F " x ) ) e. _V -> ( A. s e. ran ( x e. L |-> ( `' F " x ) ) A. t e. ran ( x e. L |-> ( `' F " x ) ) ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) <-> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) ) )
46	1, 43, 44, 45	4syl 19	. 2 |- ( ph -> ( A. s e. ran ( x e. L |-> ( `' F " x ) ) A. t e. ran ( x e. L |-> ( `' F " x ) ) ( s i^i t ) e. ran ( x e. L |-> ( `' F " x ) ) <-> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) ) )
47	42, 46	mpbid 215	1 |- ( ph -> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057   i^i cin 3415   C_ wss 3416   |-> cmpt 4475   `'ccnv 4852   ran crn 4854   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6315   ficfi 7950   fBascfbas 19007   Filcfil 20909   FilMap cfm 20997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-en 7596  df-fin 7599  df-fi 7951  df-fbas 19016  df-fil 20910

This theorem is referenced by:  fmfnfmlem4  21021