Theorem fmfnfmlem3 20908

Description: Lemma for fmfnfm 20910. (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 20806	. . . . . . . . 9 |- ( ( L e. ( Fil ` X ) /\ y e. L /\ z e. L ) -> ( y i^i z ) e. L )
3	2	3expb 1206	. . . . . . . 8 |- ( ( L e. ( Fil ` X ) /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
4	1, 3	sylan 473	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
5		fmfnfm.f	. . . . . . . . 9 |- ( ph -> F : Y --> X )
6		ffun 5686	. . . . . . . . 9 |- ( F : Y --> X -> Fun F )
7		funcnvcnv 5597	. . . . . . . . 9 |- ( Fun F -> Fun `' `' F )
8		imain 5615	. . . . . . . . . 10 |- ( Fun `' `' F -> ( `' F " ( y i^i z ) ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
9	8	eqcomd 2429	. . . . . . . . 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 466	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
12		imaeq2 5121	. . . . . . . . 9 |- ( x = ( y i^i z ) -> ( `' F " x ) = ( `' F " ( y i^i z ) ) )
13	12	eqeq2d 2433	. . . . . . . 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 3120	. . . . . . 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 665	. . . . . 6 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) )
16		ineq12 3597	. . . . . . . 8 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( s i^i t ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
17	16	eqeq1d 2425	. . . . . . 7 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( ( s i^i t ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
18	17	rexbidv 2873	. . . . . 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 225	. . . . 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 2858	. . . 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 5121	. . . . . . . 8 |- ( x = y -> ( `' F " x ) = ( `' F " y ) )
22	21	eqeq2d 2433	. . . . . . 7 |- ( x = y -> ( s = ( `' F " x ) <-> s = ( `' F " y ) ) )
23	22	cbvrexv 2992	. . . . . 6 |- ( E. x e. L s = ( `' F " x ) <-> E. y e. L s = ( `' F " y ) )
24		imaeq2 5121	. . . . . . . 8 |- ( x = z -> ( `' F " x ) = ( `' F " z ) )
25	24	eqeq2d 2433	. . . . . . 7 |- ( x = z -> ( t = ( `' F " x ) <-> t = ( `' F " z ) ) )
26	25	cbvrexv 2992	. . . . . 6 |- ( E. x e. L t = ( `' F " x ) <-> E. z e. L t = ( `' F " z ) )
27	23, 26	anbi12i 701	. . . . 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 3020	. . . . . . 7 |- s e. _V
29		eqid 2423	. . . . . . . 8 |- ( x e. L |-> ( `' F " x ) ) = ( x e. L |-> ( `' F " x ) )
30	29	elrnmpt 5038	. . . . . . 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 3020	. . . . . . 7 |- t e. _V
33	29	elrnmpt 5038	. . . . . . 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 701	. . . . 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 2930	. . . . 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 280	. . . 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 4503	. . . . 5 |- ( s i^i t ) e. _V
39	29	elrnmpt 5038	. . . . 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 273	. . 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 2780	. 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 6089	. . 3 |- ( L e. ( Fil ` X ) -> ( x e. L |-> ( `' F " x ) ) e. _V )
44		rnexg 6678	. . 3 |- ( ( x e. L |-> ( `' F " x ) ) e. _V -> ran ( x e. L |-> ( `' F " x ) ) e. _V )
45		inficl 7887	. . 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 213	1 |- ( ph -> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   A.wral 2709   E.wrex 2710   _Vcvv 3017   i^i cin 3373   C_ wss 3374   |-> cmpt 4420   `'ccnv 4790   ran crn 4792   "cima 4794   Fun wfun 5533   -->wf 5535   ` cfv 5539  (class class class)co 6244   ficfi 7872   fBascfbas 18896   Filcfil 20797   FilMap cfm 20885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7520  df-fin 7523  df-fi 7873  df-fbas 18905  df-fil 20798

This theorem is referenced by:  fmfnfmlem4  20909