Theorem fmptco 5871

Description: Composition of two functions expressed as ordered-pair class abstractions. If F has the equation ( x + 2 ) and G the equation ( 3 * z ) then ( G o. F ) has the equation ( 3 * ( x + 2 ) ). (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
fmptco.1	|- ( ( ph /\ x e. A ) -> R e. B )
fmptco.2	|- ( ph -> F = ( x e. A |-> R ) )
fmptco.3	|- ( ph -> G = ( y e. B |-> S ) )
fmptco.4	|- ( y = R -> S = T )

Assertion

Ref	Expression
fmptco	|- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Distinct variable groups:   x, A   x, y, B   y, R   ph, x   x, S   y, T

Allowed substitution hints:   ph( y)   A( y)   R( x)   S( y)   T( x)   F( x, y)   G( x, y)

Proof of Theorem fmptco

Dummy variables v u w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5331	. 2 |- Rel ( G o. F )
2		funmpt 5449	. . 3 |- Fun ( x e. A |-> T )
3		funrel 5430	. . 3 |- ( Fun ( x e. A |-> T ) -> Rel ( x e. A |-> T ) )
4	2, 3	ax-mp 5	. 2 |- Rel ( x e. A |-> T )
5		fmptco.1	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> R e. B )
6		eqid 2438	. . . . . . . . . . . . 13 |- ( x e. A |-> R ) = ( x e. A |-> R )
7	5, 6	fmptd 5862	. . . . . . . . . . . 12 |- ( ph -> ( x e. A |-> R ) : A --> B )
8		fmptco.2	. . . . . . . . . . . . 13 |- ( ph -> F = ( x e. A |-> R ) )
9	8	feq1d 5541	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> B <-> ( x e. A |-> R ) : A --> B ) )
10	7, 9	mpbird 232	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
11		ffun 5556	. . . . . . . . . . 11 |- ( F : A --> B -> Fun F )
12	10, 11	syl 16	. . . . . . . . . 10 |- ( ph -> Fun F )
13		funbrfv 5725	. . . . . . . . . . 11 |- ( Fun F -> ( z F u -> ( F ` z ) = u ) )
14	13	imp 429	. . . . . . . . . 10 |- ( ( Fun F /\ z F u ) -> ( F ` z ) = u )
15	12, 14	sylan 471	. . . . . . . . 9 |- ( ( ph /\ z F u ) -> ( F ` z ) = u )
16	15	eqcomd 2443	. . . . . . . 8 |- ( ( ph /\ z F u ) -> u = ( F ` z ) )
17	16	a1d 25	. . . . . . 7 |- ( ( ph /\ z F u ) -> ( u G w -> u = ( F ` z ) ) )
18	17	expimpd 603	. . . . . 6 |- ( ph -> ( ( z F u /\ u G w ) -> u = ( F ` z ) ) )
19	18	pm4.71rd 635	. . . . 5 |- ( ph -> ( ( z F u /\ u G w ) <-> ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
20	19	exbidv 1680	. . . 4 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
21		fvex 5696	. . . . . 6 |- ( F ` z ) e. _V
22		breq2 4291	. . . . . . 7 |- ( u = ( F ` z ) -> ( z F u <-> z F ( F ` z ) ) )
23		breq1 4290	. . . . . . 7 |- ( u = ( F ` z ) -> ( u G w <-> ( F ` z ) G w ) )
24	22, 23	anbi12d 710	. . . . . 6 |- ( u = ( F ` z ) -> ( ( z F u /\ u G w ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) ) )
25	21, 24	ceqsexv 3004	. . . . 5 |- ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) )
26		funfvbrb 5811	. . . . . . . . 9 |- ( Fun F -> ( z e. dom F <-> z F ( F ` z ) ) )
27	12, 26	syl 16	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z F ( F ` z ) ) )
28		fdm 5558	. . . . . . . . . 10 |- ( F : A --> B -> dom F = A )
29	10, 28	syl 16	. . . . . . . . 9 |- ( ph -> dom F = A )
30	29	eleq2d 2505	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z e. A ) )
31	27, 30	bitr3d 255	. . . . . . 7 |- ( ph -> ( z F ( F ` z ) <-> z e. A ) )
32	8	fveq1d 5688	. . . . . . . 8 |- ( ph -> ( F ` z ) = ( ( x e. A |-> R ) ` z ) )
33		fmptco.3	. . . . . . . 8 |- ( ph -> G = ( y e. B |-> S ) )
34		eqidd 2439	. . . . . . . 8 |- ( ph -> w = w )
35	32, 33, 34	breq123d 4301	. . . . . . 7 |- ( ph -> ( ( F ` z ) G w <-> ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) )
36	31, 35	anbi12d 710	. . . . . 6 |- ( ph -> ( ( z F ( F ` z ) /\ ( F ` z ) G w ) <-> ( z e. A /\ ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) ) )
37		nfcv 2574	. . . . . . . . 9 |- F/_ x z
38		nfv 1673	. . . . . . . . . 10 |- F/ x ph
39		nffvmpt1 5694	. . . . . . . . . . . 12 |- F/_ x ( ( x e. A |-> R ) ` z )
40		nfcv 2574	. . . . . . . . . . . 12 |- F/_ x ( y e. B |-> S )
41		nfcv 2574	. . . . . . . . . . . 12 |- F/_ x w
42	39, 40, 41	nfbr 4331	. . . . . . . . . . 11 |- F/ x ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w
43		nfcsb1v 3299	. . . . . . . . . . . 12 |- F/_ x [_ z / x ]_ T
44	43	nfeq2 2585	. . . . . . . . . . 11 |- F/ x w = [_ z / x ]_ T
45	42, 44	nfbi 1866	. . . . . . . . . 10 |- F/ x ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T )
46	38, 45	nfim 1852	. . . . . . . . 9 |- F/ x ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
47		fveq2 5686	. . . . . . . . . . . 12 |- ( x = z -> ( ( x e. A |-> R ) ` x ) = ( ( x e. A |-> R ) ` z ) )
48	47	breq1d 4297	. . . . . . . . . . 11 |- ( x = z -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) )
49		csbeq1a 3292	. . . . . . . . . . . 12 |- ( x = z -> T = [_ z / x ]_ T )
50	49	eqeq2d 2449	. . . . . . . . . . 11 |- ( x = z -> ( w = T <-> w = [_ z / x ]_ T ) )
51	48, 50	bibi12d 321	. . . . . . . . . 10 |- ( x = z -> ( ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) <-> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) )
52	51	imbi2d 316	. . . . . . . . 9 |- ( x = z -> ( ( ph -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) ) <-> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) ) )
53		vex 2970	. . . . . . . . . . . 12 |- w e. _V
54		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> y = R )
55	54	eleq1d 2504	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( y e. B <-> R e. B ) )
56		simpr 461	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> u = w )
57		fmptco.4	. . . . . . . . . . . . . . . 16 |- ( y = R -> S = T )
58	57	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> S = T )
59	56, 58	eqeq12d 2452	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( u = S <-> w = T ) )
60	55, 59	anbi12d 710	. . . . . . . . . . . . 13 |- ( ( y = R /\ u = w ) -> ( ( y e. B /\ u = S ) <-> ( R e. B /\ w = T ) ) )
61		df-mpt 4347	. . . . . . . . . . . . 13 |- ( y e. B |-> S ) = { <. y , u >. | ( y e. B /\ u = S ) }
62	60, 61	brabga 4598	. . . . . . . . . . . 12 |- ( ( R e. B /\ w e. _V ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
63	5, 53, 62	sylancl 662	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
64		simpr 461	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> x e. A )
65	6	fvmpt2 5776	. . . . . . . . . . . . 13 |- ( ( x e. A /\ R e. B ) -> ( ( x e. A |-> R ) ` x ) = R )
66	64, 5, 65	syl2anc 661	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> R ) ` x ) = R )
67	66	breq1d 4297	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> R ( y e. B |-> S ) w ) )
68	5	biantrurd 508	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( w = T <-> ( R e. B /\ w = T ) ) )
69	63, 67, 68	3bitr4d 285	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) )
70	69	expcom 435	. . . . . . . . 9 |- ( x e. A -> ( ph -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) ) )
71	37, 46, 52, 70	vtoclgaf 3030	. . . . . . . 8 |- ( z e. A -> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) )
72	71	impcom 430	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
73	72	pm5.32da 641	. . . . . 6 |- ( ph -> ( ( z e. A /\ ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
74	36, 73	bitrd 253	. . . . 5 |- ( ph -> ( ( z F ( F ` z ) /\ ( F ` z ) G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
75	25, 74	syl5bb 257	. . . 4 |- ( ph -> ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
76	20, 75	bitrd 253	. . 3 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
77		vex 2970	. . . 4 |- z e. _V
78	77, 53	opelco 5006	. . 3 |- ( <. z , w >. e. ( G o. F ) <-> E. u ( z F u /\ u G w ) )
79		df-mpt 4347	. . . . 5 |- ( x e. A |-> T ) = { <. x , v >. | ( x e. A /\ v = T ) }
80	79	eleq2i 2502	. . . 4 |- ( <. z , w >. e. ( x e. A |-> T ) <-> <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } )
81		nfv 1673	. . . . . 6 |- F/ x z e. A
82	43	nfeq2 2585	. . . . . 6 |- F/ x v = [_ z / x ]_ T
83	81, 82	nfan 1860	. . . . 5 |- F/ x ( z e. A /\ v = [_ z / x ]_ T )
84		nfv 1673	. . . . 5 |- F/ v ( z e. A /\ w = [_ z / x ]_ T )
85		eleq1 2498	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
86	49	eqeq2d 2449	. . . . . 6 |- ( x = z -> ( v = T <-> v = [_ z / x ]_ T ) )
87	85, 86	anbi12d 710	. . . . 5 |- ( x = z -> ( ( x e. A /\ v = T ) <-> ( z e. A /\ v = [_ z / x ]_ T ) ) )
88		eqeq1 2444	. . . . . 6 |- ( v = w -> ( v = [_ z / x ]_ T <-> w = [_ z / x ]_ T ) )
89	88	anbi2d 703	. . . . 5 |- ( v = w -> ( ( z e. A /\ v = [_ z / x ]_ T ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
90	83, 84, 77, 53, 87, 89	opelopabf 4608	. . . 4 |- ( <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } <-> ( z e. A /\ w = [_ z / x ]_ T ) )
91	80, 90	bitri 249	. . 3 |- ( <. z , w >. e. ( x e. A |-> T ) <-> ( z e. A /\ w = [_ z / x ]_ T ) )
92	76, 78, 91	3bitr4g 288	. 2 |- ( ph -> ( <. z , w >. e. ( G o. F ) <-> <. z , w >. e. ( x e. A |-> T ) ) )
93	1, 4, 92	eqrelrdv 4931	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   _Vcvv 2967   [_csb 3283   <.cop 3878   class class class wbr 4287   {copab 4344   e. cmpt 4345   dom cdm 4835   o. ccom 4839   Rel wrel 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421

This theorem is referenced by:  fmptcof  5872  fcompt  5874  fcoconst  5875  ofco  6335  ccatco  12455  lo1o12  13003  rlimcn1  13058  rlimcn1b  13059  rlimdiv  13115  ackbijnn  13283  setcepi  14948  prf1st  15006  prf2nd  15007  hofcllem  15060  prdsidlem  15445  pws0g  15449  pwsco1mhm  15489  pwsco2mhm  15490  pwsinvg  15658  pwssub  15659  galactghm  15899  efginvrel1  16216  frgpup3lem  16265  gsumzf1o  16382  gsumzf1oOLD  16385  gsumconst  16417  gsumzmhm  16420  gsumzmhmOLD  16421  gsummhm2  16424  gsummhm2OLD  16425  gsummptmhm  16426  gsumsub  16437  gsumsubOLD  16438  gsum2dlem2  16450  gsum2dOLD  16452  dprdfsub  16499  dprdfsubOLD  16506  lmhmvsca  17103  psrass1lem  17424  psrlinv  17445  psrcom  17458  evlslem2  17572  coe1fval3  17639  psropprmul  17668  coe1z  17692  coe1mul2  17698  coe1tm  17701  ply1coe  17721  ply1coeOLD  17722  evls1sca  17733  frgpcyg  17981  evpmodpmf1o  18001  mhmvlin  18272  ofco2  18307  mdetleib2  18374  mdetralt  18389  smadiadetlem3  18449  ptrescn  19187  lmcn2  19197  qtopeu  19264  flfcnp2  19555  tgpconcomp  19658  tsmsmhm  19695  tsmssub  19698  tsmsxplem1  19702  negfcncf  20470  pcopt  20569  pcopt2  20570  pi1xfrcnvlem  20603  ovolctb  20948  ovolfs2  21026  uniioombllem2  21038  uniioombllem3  21040  ismbf  21083  mbfconst  21088  ismbfcn2  21092  itg1climres  21167  iblabslem  21280  iblabs  21281  bddmulibl  21291  limccnp  21341  limccnp2  21342  limcco  21343  dvcof  21397  dvcjbr  21398  dvcj  21399  dvfre  21400  dvmptcj  21417  dvmptco  21421  dvcnvlem  21423  dvef  21427  dvlip  21440  dvlipcn  21441  itgsubstlem  21495  plypf1  21655  plyco  21684  dgrcolem1  21715  dgrcolem2  21716  dgrco  21717  plycjlem  21718  taylply2  21808  logcn  22067  leibpi  22312  efrlim  22338  jensenlem2  22356  amgmlem  22358  ftalem7  22391  lgseisenlem4  22666  dchrisum0  22744  cofmpt  25932  ofcfval4  26499  eulerpartgbij  26707  dstfrvclim1  26812  lgamgulmlem2  26968  lgamcvg2  26993  cvmliftlem6  27131  cvmliftphtlem  27158  cvmlift3lem5  27164  circum  27270  mblfinlem2  28382  volsupnfl  28389  itgaddnc  28405  itgmulc2nc  28413  ftc1anclem1  28420  ftc1anclem2  28421  ftc1anclem3  28422  ftc1anclem4  28423  ftc1anclem5  28424  ftc1anclem6  28425  ftc1anclem7  28426  ftc1anclem8  28427  fnopabco  28569  upixp  28576  mendassa  29504