Theorem fmfnfmlem3 20192

Description: Lemma for fmfnfm 20194. (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 20090	. . . . . . . . 9 |- ( ( L e. ( Fil ` X ) /\ y e. L /\ z e. L ) -> ( y i^i z ) e. L )
3	2	3expb 1197	. . . . . . . 8 |- ( ( L e. ( Fil ` X ) /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
4	1, 3	sylan 471	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
5		fmfnfm.f	. . . . . . . . 9 |- ( ph -> F : Y --> X )
6		ffun 5731	. . . . . . . . 9 |- ( F : Y --> X -> Fun F )
7		funcnvcnv 5644	. . . . . . . . 9 |- ( Fun F -> Fun `' `' F )
8		imain 5662	. . . . . . . . . 10 |- ( Fun `' `' F -> ( `' F " ( y i^i z ) ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
9	8	eqcomd 2475	. . . . . . . . 9 |- ( Fun `' `' F -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
10	5, 6, 7, 9	4syl 21	. . . . . . . 8 |- ( ph -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
11	10	adantr 465	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
12		imaeq2 5331	. . . . . . . . 9 |- ( x = ( y i^i z ) -> ( `' F " x ) = ( `' F " ( y i^i z ) ) )
13	12	eqeq2d 2481	. . . . . . . 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 3214	. . . . . . 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 661	. . . . . 6 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) )
16		ineq12 3695	. . . . . . . 8 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( s i^i t ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
17	16	eqeq1d 2469	. . . . . . 7 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( ( s i^i t ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
18	17	rexbidv 2973	. . . . . 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 222	. . . . 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 2962	. . . 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 5331	. . . . . . . 8 |- ( x = y -> ( `' F " x ) = ( `' F " y ) )
22	21	eqeq2d 2481	. . . . . . 7 |- ( x = y -> ( s = ( `' F " x ) <-> s = ( `' F " y ) ) )
23	22	cbvrexv 3089	. . . . . 6 |- ( E. x e. L s = ( `' F " x ) <-> E. y e. L s = ( `' F " y ) )
24		imaeq2 5331	. . . . . . . 8 |- ( x = z -> ( `' F " x ) = ( `' F " z ) )
25	24	eqeq2d 2481	. . . . . . 7 |- ( x = z -> ( t = ( `' F " x ) <-> t = ( `' F " z ) ) )
26	25	cbvrexv 3089	. . . . . 6 |- ( E. x e. L t = ( `' F " x ) <-> E. z e. L t = ( `' F " z ) )
27	23, 26	anbi12i 697	. . . . 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 3116	. . . . . . 7 |- s e. _V
29		eqid 2467	. . . . . . . 8 |- ( x e. L |-> ( `' F " x ) ) = ( x e. L |-> ( `' F " x ) )
30	29	elrnmpt 5247	. . . . . . 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 3116	. . . . . . 7 |- t e. _V
33	29	elrnmpt 5247	. . . . . . 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 697	. . . . 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 3029	. . . . 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 277	. . . 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 4588	. . . . 5 |- ( s i^i t ) e. _V
39	29	elrnmpt 5247	. . . . 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 270	. . 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 2884	. 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 6128	. . 3 |- ( L e. ( Fil ` X ) -> ( x e. L |-> ( `' F " x ) ) e. _V )
44		rnexg 6713	. . 3 |- ( ( x e. L |-> ( `' F " x ) ) e. _V -> ran ( x e. L |-> ( `' F " x ) ) e. _V )
45		inficl 7881	. . 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 21	. 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 210	1 |- ( ph -> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   ficfi 7866   fBascfbas 18177   Filcfil 20081   FilMap cfm 20169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-fbas 18187  df-fil 20082

This theorem is referenced by:  fmfnfmlem4  20193