Theorem fmfnfmlem3 20582

Description: Lemma for fmfnfm 20584. (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 20480	. . . . . . . . 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 5739	. . . . . . . . 9 |- ( F : Y --> X -> Fun F )
7		funcnvcnv 5652	. . . . . . . . 9 |- ( Fun F -> Fun `' `' F )
8		imain 5670	. . . . . . . . . 10 |- ( Fun `' `' F -> ( `' F " ( y i^i z ) ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
9	8	eqcomd 2465	. . . . . . . . 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 5343	. . . . . . . . 9 |- ( x = ( y i^i z ) -> ( `' F " x ) = ( `' F " ( y i^i z ) ) )
13	12	eqeq2d 2471	. . . . . . . 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 3210	. . . . . . 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 3691	. . . . . . . 8 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( s i^i t ) = ( ( `' F " y ) i^i ( `' F " z ) ) )
17	16	eqeq1d 2459	. . . . . . 7 |- ( ( s = ( `' F " y ) /\ t = ( `' F " z ) ) -> ( ( s i^i t ) = ( `' F " x ) <-> ( ( `' F " y ) i^i ( `' F " z ) ) = ( `' F " x ) ) )
18	17	rexbidv 2968	. . . . . 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 2956	. . . 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 5343	. . . . . . . 8 |- ( x = y -> ( `' F " x ) = ( `' F " y ) )
22	21	eqeq2d 2471	. . . . . . 7 |- ( x = y -> ( s = ( `' F " x ) <-> s = ( `' F " y ) ) )
23	22	cbvrexv 3085	. . . . . 6 |- ( E. x e. L s = ( `' F " x ) <-> E. y e. L s = ( `' F " y ) )
24		imaeq2 5343	. . . . . . . 8 |- ( x = z -> ( `' F " x ) = ( `' F " z ) )
25	24	eqeq2d 2471	. . . . . . 7 |- ( x = z -> ( t = ( `' F " x ) <-> t = ( `' F " z ) ) )
26	25	cbvrexv 3085	. . . . . 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 3112	. . . . . . 7 |- s e. _V
29		eqid 2457	. . . . . . . 8 |- ( x e. L |-> ( `' F " x ) ) = ( x e. L |-> ( `' F " x ) )
30	29	elrnmpt 5259	. . . . . . 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 3112	. . . . . . 7 |- t e. _V
33	29	elrnmpt 5259	. . . . . . 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 3025	. . . . 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 4597	. . . . 5 |- ( s i^i t ) e. _V
39	29	elrnmpt 5259	. . . . 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 2877	. 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 6143	. . 3 |- ( L e. ( Fil ` X ) -> ( x e. L |-> ( `' F " x ) ) e. _V )
44		rnexg 6731	. . 3 |- ( ( x e. L |-> ( `' F " x ) ) e. _V -> ran ( x e. L |-> ( `' F " x ) ) e. _V )
45		inficl 7903	. . 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 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   |-> cmpt 4515   `'ccnv 5007   ran crn 5009   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   ficfi 7888   fBascfbas 18532   Filcfil 20471   FilMap cfm 20559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-fin 7539  df-fi 7889  df-fbas 18542  df-fil 20472

This theorem is referenced by:  fmfnfmlem4  20583