Theorem offval2 6914

Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval2.1	|- ( ph -> A e. V )
offval2.2	|- ( ( ph /\ x e. A ) -> B e. W )
offval2.3	|- ( ( ph /\ x e. A ) -> C e. X )
offval2.4	|- ( ph -> F = ( x e. A |-> B ) )
offval2.5	|- ( ph -> G = ( x e. A |-> C ) )

Assertion

Ref	Expression
offval2	|- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Distinct variable groups:   x, A   ph, x   x, R

Allowed substitution hints:   B( x)   C( x)   F( x)   G( x)   V( x)   W( x)   X( x)

Proof of Theorem offval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2.2	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. W )
2	1	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. A B e. W )
3		eqid 2622	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 6020	. . . . 5 |- ( A. x e. A B e. W -> ( x e. A |-> B ) Fn A )
5	2, 4	syl 17	. . . 4 |- ( ph -> ( x e. A |-> B ) Fn A )
6		offval2.4	. . . . 5 |- ( ph -> F = ( x e. A |-> B ) )
7	6	fneq1d 5981	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 247	. . 3 |- ( ph -> F Fn A )
9		offval2.3	. . . . . 6 |- ( ( ph /\ x e. A ) -> C e. X )
10	9	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. A C e. X )
11		eqid 2622	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
12	11	fnmpt 6020	. . . . 5 |- ( A. x e. A C e. X -> ( x e. A |-> C ) Fn A )
13	10, 12	syl 17	. . . 4 |- ( ph -> ( x e. A |-> C ) Fn A )
14		offval2.5	. . . . 5 |- ( ph -> G = ( x e. A |-> C ) )
15	14	fneq1d 5981	. . . 4 |- ( ph -> ( G Fn A <-> ( x e. A |-> C ) Fn A ) )
16	13, 15	mpbird 247	. . 3 |- ( ph -> G Fn A )
17		offval2.1	. . 3 |- ( ph -> A e. V )
18		inidm 3822	. . 3 |- ( A i^i A ) = A
19	6	adantr 481	. . . 4 |- ( ( ph /\ y e. A ) -> F = ( x e. A |-> B ) )
20	19	fveq1d 6193	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( ( x e. A |-> B ) ` y ) )
21	14	adantr 481	. . . 4 |- ( ( ph /\ y e. A ) -> G = ( x e. A |-> C ) )
22	21	fveq1d 6193	. . 3 |- ( ( ph /\ y e. A ) -> ( G ` y ) = ( ( x e. A |-> C ) ` y ) )
23	8, 16, 17, 17, 18, 20, 22	offval 6904	. 2 |- ( ph -> ( F oF R G ) = ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) )
24		nffvmpt1 6199	. . . . 5 |- F/_ x ( ( x e. A |-> B ) ` y )
25		nfcv 2764	. . . . 5 |- F/_ x R
26		nffvmpt1 6199	. . . . 5 |- F/_ x ( ( x e. A |-> C ) ` y )
27	24, 25, 26	nfov 6676	. . . 4 |- F/_ x ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) )
28		nfcv 2764	. . . 4 |- F/_ y ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) )
29		fveq2 6191	. . . . 5 |- ( y = x -> ( ( x e. A |-> B ) ` y ) = ( ( x e. A |-> B ) ` x ) )
30		fveq2 6191	. . . . 5 |- ( y = x -> ( ( x e. A |-> C ) ` y ) = ( ( x e. A |-> C ) ` x ) )
31	29, 30	oveq12d 6668	. . . 4 |- ( y = x -> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) = ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
32	27, 28, 31	cbvmpt 4749	. . 3 |- ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) )
33		simpr 477	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
34	3	fvmpt2 6291	. . . . . 6 |- ( ( x e. A /\ B e. W ) -> ( ( x e. A |-> B ) ` x ) = B )
35	33, 1, 34	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
36	11	fvmpt2 6291	. . . . . 6 |- ( ( x e. A /\ C e. X ) -> ( ( x e. A |-> C ) ` x ) = C )
37	33, 9, 36	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> C ) ` x ) = C )
38	35, 37	oveq12d 6668	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) = ( B R C ) )
39	38	mpteq2dva 4744	. . 3 |- ( ph -> ( x e. A |-> ( ( ( x e. A |-> B ) ` x ) R ( ( x e. A |-> C ) ` x ) ) ) = ( x e. A |-> ( B R C ) ) )
40	32, 39	syl5eq 2668	. 2 |- ( ph -> ( y e. A |-> ( ( ( x e. A |-> B ) ` y ) R ( ( x e. A |-> C ) ` y ) ) ) = ( x e. A |-> ( B R C ) ) )
41	23, 40	eqtrd 2656	1 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-of 6897

This theorem is referenced by:  ofmpteq  6916  ofc12  6922  caofinvl  6924  caofcom  6929  caofass  6931  caofdi  6933  caofdir  6934  caonncan  6935  offval22  7253  ofccat  13708  ofs1  13709  o1add2  14354  o1mul2  14355  o1sub2  14356  o1dif  14360  fsumo1  14544  pwsplusgval  16150  pwsmulrval  16151  pwsvscafval  16154  pwsco1mhm  17370  pwsco2mhm  17371  pwssub  17529  gsumzaddlem  18321  gsummptfsadd  18324  gsummptfidmadd2  18326  gsumzsplit  18327  gsumsub  18348  gsummptfssub  18349  dprdfadd  18419  dprdfsub  18420  dprdfeq0  18421  dprdf11  18422  lmhmvsca  19045  rrgsupp  19291  psrbagaddcl  19370  psrass1lem  19377  psrlinv  19397  psrass1  19405  psrdi  19406  psrdir  19407  psrass23l  19408  psrcom  19409  psrass23  19410  mplsubrglem  19439  mplmonmul  19464  mplcoe1  19465  mplcoe3  19466  mplcoe5  19468  mplmon2  19493  evlslem1  19515  coe1sclmul  19652  coe1sclmul2  19654  uvcresum  20132  grpvrinv  20202  mhmvlin  20203  mamudi  20209  mamudir  20210  mdetunilem9  20426  tsmssub  21952  tgptsmscls  21953  tsmssplit  21955  tsmsxplem2  21957  ovolctb  23258  mbfmulc2re  23415  mbfneg  23417  mbfadd  23428  mbfsub  23429  mbfmulc2  23430  mbfmul  23493  itg2const  23507  itg2mulclem  23513  itg2mulc  23514  itg2splitlem  23515  itg2monolem1  23517  i1fibl  23574  itgitg1  23575  ibladdlem  23586  ibladd  23587  itgaddlem1  23589  iblabslem  23594  iblabs  23595  iblmulc2  23597  itgmulc2lem1  23598  bddmulibl  23605  dvmulf  23706  dvcmulf  23708  dvcof  23711  dvexp  23716  dvmptadd  23723  dvmptmul  23724  dvmptco  23735  dvef  23743  dv11cn  23764  itgsubstlem  23811  mdegmullem  23838  plypf1  23968  plyaddlem1  23969  plymullem1  23970  plyco  23997  dgrcolem1  24029  dgrcolem2  24030  plydiveu  24053  plyremlem  24059  elqaalem3  24076  iaa  24080  taylply2  24122  ulmdvlem1  24154  iblulm  24161  jensenlem2  24714  amgmlem  24716  ftalem7  24805  basellem8  24814  basellem9  24815  dchrmulid2  24977  dchrinvcl  24978  dchrfi  24980  lgseisenlem3  25102  lgseisenlem4  25103  chtppilimlem2  25163  chebbnd2  25166  chto1lb  25167  chpchtlim  25168  chpo1ub  25169  vmadivsum  25171  rpvmasumlem  25176  mudivsum  25219  selberglem1  25234  selberglem2  25235  selberg2lem  25239  selberg2  25240  pntrsumo1  25254  selbergr  25257  ofoprabco  29464  pl1cn  30001  esumadd  30119  poimirlem16  33425  poimirlem19  33428  itg2addnclem  33461  itg2addnclem3  33463  ibladdnclem  33466  itgaddnclem1  33468  iblabsnclem  33473  iblabsnc  33474  iblmulc2nc  33475  itgmulc2nclem1  33476  itgmulc2nclem2  33477  itgmulc2nc  33478  itgabsnc  33479  ftc1anclem3  33487  ftc1anclem4  33488  ftc1anclem5  33489  ftc1anclem6  33490  ftc1anclem7  33491  ftc1anclem8  33492  mendlmod  37763  mendassa  37764  expgrowthi  38532  expgrowth  38534  binomcxplemrat  38549  mulcncff  40081  subcncff  40093  addcncff  40097  divcncff  40104  dvsubf  40128  dvdivf  40137  fourierdlem16  40340  fourierdlem21  40345  fourierdlem22  40346  fourierdlem58  40381  fourierdlem59  40382  fourierdlem72  40395  fourierdlem83  40406  offvalfv  42121  aacllem  42547  amgmwlem  42548  amgmlemALT  42549