Theorem fmptco 6072

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 5340	. 2 |- Rel ( G o. F )
2		funmpt 5625	. . 3 |- Fun ( x e. A |-> T )
3		funrel 5606	. . 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 2471	. . . . . . . . . . . . 13 |- ( x e. A |-> R ) = ( x e. A |-> R )
7	5, 6	fmptd 6061	. . . . . . . . . . . 12 |- ( ph -> ( x e. A |-> R ) : A --> B )
8		fmptco.2	. . . . . . . . . . . . 13 |- ( ph -> F = ( x e. A |-> R ) )
9	8	feq1d 5724	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> B <-> ( x e. A |-> R ) : A --> B ) )
10	7, 9	mpbird 240	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
11		ffun 5742	. . . . . . . . . . 11 |- ( F : A --> B -> Fun F )
12	10, 11	syl 17	. . . . . . . . . 10 |- ( ph -> Fun F )
13		funbrfv 5917	. . . . . . . . . . 11 |- ( Fun F -> ( z F u -> ( F ` z ) = u ) )
14	13	imp 436	. . . . . . . . . 10 |- ( ( Fun F /\ z F u ) -> ( F ` z ) = u )
15	12, 14	sylan 479	. . . . . . . . 9 |- ( ( ph /\ z F u ) -> ( F ` z ) = u )
16	15	eqcomd 2477	. . . . . . . 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 614	. . . . . 6 |- ( ph -> ( ( z F u /\ u G w ) -> u = ( F ` z ) ) )
19	18	pm4.71rd 647	. . . . 5 |- ( ph -> ( ( z F u /\ u G w ) <-> ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
20	19	exbidv 1776	. . . 4 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
21		fvex 5889	. . . . . 6 |- ( F ` z ) e. _V
22		breq2 4399	. . . . . . 7 |- ( u = ( F ` z ) -> ( z F u <-> z F ( F ` z ) ) )
23		breq1 4398	. . . . . . 7 |- ( u = ( F ` z ) -> ( u G w <-> ( F ` z ) G w ) )
24	22, 23	anbi12d 725	. . . . . 6 |- ( u = ( F ` z ) -> ( ( z F u /\ u G w ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) ) )
25	21, 24	ceqsexv 3070	. . . . 5 |- ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) )
26		funfvbrb 6010	. . . . . . . . 9 |- ( Fun F -> ( z e. dom F <-> z F ( F ` z ) ) )
27	12, 26	syl 17	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z F ( F ` z ) ) )
28		fdm 5745	. . . . . . . . . 10 |- ( F : A --> B -> dom F = A )
29	10, 28	syl 17	. . . . . . . . 9 |- ( ph -> dom F = A )
30	29	eleq2d 2534	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z e. A ) )
31	27, 30	bitr3d 263	. . . . . . 7 |- ( ph -> ( z F ( F ` z ) <-> z e. A ) )
32	8	fveq1d 5881	. . . . . . . 8 |- ( ph -> ( F ` z ) = ( ( x e. A |-> R ) ` z ) )
33		fmptco.3	. . . . . . . 8 |- ( ph -> G = ( y e. B |-> S ) )
34		eqidd 2472	. . . . . . . 8 |- ( ph -> w = w )
35	32, 33, 34	breq123d 4409	. . . . . . 7 |- ( ph -> ( ( F ` z ) G w <-> ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) )
36	31, 35	anbi12d 725	. . . . . 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 2612	. . . . . . . . 9 |- F/_ x z
38		nfv 1769	. . . . . . . . . 10 |- F/ x ph
39		nffvmpt1 5887	. . . . . . . . . . . 12 |- F/_ x ( ( x e. A |-> R ) ` z )
40		nfcv 2612	. . . . . . . . . . . 12 |- F/_ x ( y e. B |-> S )
41		nfcv 2612	. . . . . . . . . . . 12 |- F/_ x w
42	39, 40, 41	nfbr 4440	. . . . . . . . . . 11 |- F/ x ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w
43		nfcsb1v 3365	. . . . . . . . . . . 12 |- F/_ x [_ z / x ]_ T
44	43	nfeq2 2627	. . . . . . . . . . 11 |- F/ x w = [_ z / x ]_ T
45	42, 44	nfbi 2037	. . . . . . . . . 10 |- F/ x ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T )
46	38, 45	nfim 2023	. . . . . . . . 9 |- F/ x ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
47		fveq2 5879	. . . . . . . . . . . 12 |- ( x = z -> ( ( x e. A |-> R ) ` x ) = ( ( x e. A |-> R ) ` z ) )
48	47	breq1d 4405	. . . . . . . . . . 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 3358	. . . . . . . . . . . 12 |- ( x = z -> T = [_ z / x ]_ T )
50	49	eqeq2d 2481	. . . . . . . . . . 11 |- ( x = z -> ( w = T <-> w = [_ z / x ]_ T ) )
51	48, 50	bibi12d 328	. . . . . . . . . 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 323	. . . . . . . . 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 3034	. . . . . . . . . . . 12 |- w e. _V
54		simpl 464	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> y = R )
55	54	eleq1d 2533	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( y e. B <-> R e. B ) )
56		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> u = w )
57		fmptco.4	. . . . . . . . . . . . . . . 16 |- ( y = R -> S = T )
58	57	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> S = T )
59	56, 58	eqeq12d 2486	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( u = S <-> w = T ) )
60	55, 59	anbi12d 725	. . . . . . . . . . . . 13 |- ( ( y = R /\ u = w ) -> ( ( y e. B /\ u = S ) <-> ( R e. B /\ w = T ) ) )
61		df-mpt 4456	. . . . . . . . . . . . 13 |- ( y e. B |-> S ) = { <. y , u >. | ( y e. B /\ u = S ) }
62	60, 61	brabga 4715	. . . . . . . . . . . 12 |- ( ( R e. B /\ w e. _V ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
63	5, 53, 62	sylancl 675	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
64		simpr 468	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> x e. A )
65	6	fvmpt2 5972	. . . . . . . . . . . . 13 |- ( ( x e. A /\ R e. B ) -> ( ( x e. A |-> R ) ` x ) = R )
66	64, 5, 65	syl2anc 673	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> R ) ` x ) = R )
67	66	breq1d 4405	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> R ( y e. B |-> S ) w ) )
68	5	biantrurd 516	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( w = T <-> ( R e. B /\ w = T ) ) )
69	63, 67, 68	3bitr4d 293	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) )
70	69	expcom 442	. . . . . . . . 9 |- ( x e. A -> ( ph -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) ) )
71	37, 46, 52, 70	vtoclgaf 3098	. . . . . . . 8 |- ( z e. A -> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) )
72	71	impcom 437	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
73	72	pm5.32da 653	. . . . . 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 261	. . . . 5 |- ( ph -> ( ( z F ( F ` z ) /\ ( F ` z ) G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
75	25, 74	syl5bb 265	. . . 4 |- ( ph -> ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
76	20, 75	bitrd 261	. . 3 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
77		vex 3034	. . . 4 |- z e. _V
78	77, 53	opelco 5011	. . 3 |- ( <. z , w >. e. ( G o. F ) <-> E. u ( z F u /\ u G w ) )
79		df-mpt 4456	. . . . 5 |- ( x e. A |-> T ) = { <. x , v >. | ( x e. A /\ v = T ) }
80	79	eleq2i 2541	. . . 4 |- ( <. z , w >. e. ( x e. A |-> T ) <-> <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } )
81		nfv 1769	. . . . . 6 |- F/ x z e. A
82	43	nfeq2 2627	. . . . . 6 |- F/ x v = [_ z / x ]_ T
83	81, 82	nfan 2031	. . . . 5 |- F/ x ( z e. A /\ v = [_ z / x ]_ T )
84		nfv 1769	. . . . 5 |- F/ v ( z e. A /\ w = [_ z / x ]_ T )
85		eleq1 2537	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
86	49	eqeq2d 2481	. . . . . 6 |- ( x = z -> ( v = T <-> v = [_ z / x ]_ T ) )
87	85, 86	anbi12d 725	. . . . 5 |- ( x = z -> ( ( x e. A /\ v = T ) <-> ( z e. A /\ v = [_ z / x ]_ T ) ) )
88		eqeq1 2475	. . . . . 6 |- ( v = w -> ( v = [_ z / x ]_ T <-> w = [_ z / x ]_ T ) )
89	88	anbi2d 718	. . . . 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 4726	. . . 4 |- ( <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } <-> ( z e. A /\ w = [_ z / x ]_ T ) )
91	80, 90	bitri 257	. . 3 |- ( <. z , w >. e. ( x e. A |-> T ) <-> ( z e. A /\ w = [_ z / x ]_ T ) )
92	76, 78, 91	3bitr4g 296	. 2 |- ( ph -> ( <. z , w >. e. ( G o. F ) <-> <. z , w >. e. ( x e. A |-> T ) ) )
93	1, 4, 92	eqrelrdv 4936	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   [_csb 3349   <.cop 3965   class class class wbr 4395   {copab 4453   |-> cmpt 4454   dom cdm 4839   o. ccom 4843   Rel wrel 4844   Fun wfun 5583   -->wf 5585   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597

This theorem is referenced by:  fmptcof  6073  fcompt  6075  fcoconst  6076  ofco  6570  ccatco  12991  lo1o12  13674  rlimcn1  13729  rlimcn1b  13730  rlimdiv  13786  ackbijnn  13963  setcepi  16061  prf1st  16167  prf2nd  16168  hofcllem  16221  prdsidlem  16646  pws0g  16650  pwsco1mhm  16695  pwsco2mhm  16696  pwsinvg  16876  pwssub  16877  galactghm  17122  efginvrel1  17456  frgpup3lem  17505  gsumzf1o  17624  gsumconst  17645  gsummptshft  17647  gsumzmhm  17648  gsummhm2  17650  gsummptmhm  17651  gsumsub  17659  gsum2dlem2  17681  dprdfsub  17732  lmhmvsca  18346  psrass1lem  18678  psrlinv  18698  psrcom  18710  evlslem2  18812  coe1fval3  18878  psropprmul  18908  coe1z  18933  coe1mul2  18939  coe1tm  18943  ply1coe  18966  ply1coeOLD  18967  evls1sca  18989  frgpcyg  19221  evpmodpmf1o  19241  mhmvlin  19499  ofco2  19553  mdetleib2  19690  mdetralt  19710  smadiadetlem3  19770  ptrescn  20731  lmcn2  20741  qtopeu  20808  flfcnp2  21100  tgpconcomp  21205  tsmsmhm  21238  tsmssub  21241  tsmsxplem1  21245  negfcncf  22029  pcopt  22131  pcopt2  22132  pi1xfrcnvlem  22165  ovolctb  22521  ovolfs2  22602  uniioombllem2  22619  uniioombllem2OLD  22620  uniioombllem3  22622  ismbf  22665  mbfconst  22670  ismbfcn2  22674  itg1climres  22751  iblabslem  22864  iblabs  22865  bddmulibl  22875  limccnp  22925  limccnp2  22926  limcco  22927  dvcof  22981  dvcjbr  22982  dvcj  22983  dvfre  22984  dvmptcj  23001  dvmptco  23005  dvcnvlem  23007  dvef  23011  dvlip  23024  dvlipcn  23025  itgsubstlem  23079  plypf1  23245  plyco  23274  dgrcolem1  23306  dgrcolem2  23307  dgrco  23308  plycjlem  23309  taylply2  23402  logcn  23671  leibpi  23947  efrlim  23974  jensenlem2  23992  amgmlem  23994  lgamgulmlem2  24034  lgamcvg2  24059  ftalem7  24084  lgseisenlem4  24359  dchrisum0  24437  cofmpt  28341  ofcfval4  29000  eulerpartgbij  29278  dstfrvclim1  29383  cvmliftlem6  30085  cvmliftphtlem  30112  cvmlift3lem5  30118  elmsubrn  30238  msubco  30241  circum  30390  mblfinlem2  32042  volsupnfl  32049  itgaddnc  32066  itgmulc2nc  32074  ftc1anclem1  32081  ftc1anclem2  32082  ftc1anclem3  32083  ftc1anclem4  32084  ftc1anclem5  32085  ftc1anclem6  32086  ftc1anclem7  32087  ftc1anclem8  32088  fnopabco  32113  upixp  32120  mendassa  36131  cncfcompt  37857  dvcosax  37895  dirkercncflem4  38080  fourierdlem111  38193  meadjiunlem  38419  meadjiun  38420