Theorem fmfnfmlem3 21760

Description: Lemma for fmfnfm 21762. (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 21658	. . . . . . . . 9 |- ( ( L e. ( Fil ` X ) /\ y e. L /\ z e. L ) -> ( y i^i z ) e. L )
3	2	3expb 1266	. . . . . . . 8 |- ( ( L e. ( Fil ` X ) /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
4	1, 3	sylan 488	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( y i^i z ) e. L )
5		fmfnfm.f	. . . . . . . . 9 |- ( ph -> F : Y --> X )
6		ffun 6048	. . . . . . . . 9 |- ( F : Y --> X -> Fun F )
7		funcnvcnv 5956	. . . . . . . . 9 |- ( Fun F -> Fun `' `' F )
8		imain 5974	. . . . . . . . . 10 |- ( Fun `' `' F -> ( `' F " ( y i^i z ) ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
9	8	eqcomd 2628	. . . . . . . . 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 481	. . . . . . 7 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " ( y i^i z ) ) )
12		imaeq2 5462	. . . . . . . . 9 |- ( x = ( y i^i z ) -> ( `' F " x ) = ( `' F " ( y i^i z ) ) )
13	12	eqeq2d 2632	. . . . . . . 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 3309	. . . . . . 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 693	. . . . . 6 |- ( ( ph /\ ( y e. L /\ z e. L ) ) -> E. x e. L ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) )
16		ineq12 3809	. . . . . . . 8 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( s i^i t ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
17	16	eqeq1d 2624	. . . . . . 7 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( ( s i^i t ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
18	17	rexbidv 3052	. . . . . 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 237	. . . . 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 3038	. . . 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 5462	. . . . . . . 8 |- ( x = y -> ( `' F " x ) = ( `' F " y ) )
22	21	eqeq2d 2632	. . . . . . 7 |- ( x = y -> ( s = ( `' F " x ) <-> s = ( `' F " y ) ) )
23	22	cbvrexv 3172	. . . . . 6 |- ( E. x e. L s = ( `' F " x ) <-> E. y e. L s = ( `' F " y ) )
24		imaeq2 5462	. . . . . . . 8 |- ( x = z -> ( `' F " x ) = ( `' F " z ) )
25	24	eqeq2d 2632	. . . . . . 7 |- ( x = z -> ( t = ( `' F " x ) <-> t = ( `' F " z ) ) )
26	25	cbvrexv 3172	. . . . . 6 |- ( E. x e. L t = ( `' F " x ) <-> E. z e. L t = ( `' F " z ) )
27	23, 26	anbi12i 733	. . . . 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 3203	. . . . . . 7 |- s e. _V
29		eqid 2622	. . . . . . . 8 |- ( x e. L |-> ( `' F " x ) ) = ( x e. L |-> ( `' F " x ) )
30	29	elrnmpt 5372	. . . . . . 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 3203	. . . . . . 7 |- t e. _V
33	29	elrnmpt 5372	. . . . . . 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 733	. . . . 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 3107	. . . . 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 292	. . . 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 4799	. . . . 5 |- ( s i^i t ) e. _V
39	29	elrnmpt 5372	. . . . 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 285	. . 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 2970	. 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 6484	. . 3 |- ( L e. ( Fil ` X ) -> ( x e. L |-> ( `' F " x ) ) e. _V )
44		rnexg 7098	. . 3 |- ( ( x e. L |-> ( `' F " x ) ) e. _V -> ran ( x e. L |-> ( `' F " x ) ) e. _V )
45		inficl 8331	. . 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 222	1 |- ( ph -> ( fi ` ran ( x e. L |-> ( `' F " x ) ) ) = ran ( x e. L |-> ( `' F " x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ficfi 8316   fBascfbas 19734   Filcfil 21649   FilMap cfm 21737

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

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743  df-fil 21650

This theorem is referenced by:  fmfnfmlem4  21761