Theorem fmptco 6052

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 5503	. 2 |- Rel ( G o. F )
2		funmpt 5622	. . 3 |- Fun ( x e. A |-> T )
3		funrel 5603	. . 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 2467	. . . . . . . . . . . . 13 |- ( x e. A |-> R ) = ( x e. A |-> R )
7	5, 6	fmptd 6043	. . . . . . . . . . . 12 |- ( ph -> ( x e. A |-> R ) : A --> B )
8		fmptco.2	. . . . . . . . . . . . 13 |- ( ph -> F = ( x e. A |-> R ) )
9	8	feq1d 5715	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> B <-> ( x e. A |-> R ) : A --> B ) )
10	7, 9	mpbird 232	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
11		ffun 5731	. . . . . . . . . . 11 |- ( F : A --> B -> Fun F )
12	10, 11	syl 16	. . . . . . . . . 10 |- ( ph -> Fun F )
13		funbrfv 5904	. . . . . . . . . . 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 2475	. . . . . . . 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 1690	. . . 4 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
21		fvex 5874	. . . . . 6 |- ( F ` z ) e. _V
22		breq2 4451	. . . . . . 7 |- ( u = ( F ` z ) -> ( z F u <-> z F ( F ` z ) ) )
23		breq1 4450	. . . . . . 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 3150	. . . . 5 |- ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) )
26		funfvbrb 5992	. . . . . . . . 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 5733	. . . . . . . . . 10 |- ( F : A --> B -> dom F = A )
29	10, 28	syl 16	. . . . . . . . 9 |- ( ph -> dom F = A )
30	29	eleq2d 2537	. . . . . . . 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 5866	. . . . . . . 8 |- ( ph -> ( F ` z ) = ( ( x e. A |-> R ) ` z ) )
33		fmptco.3	. . . . . . . 8 |- ( ph -> G = ( y e. B |-> S ) )
34		eqidd 2468	. . . . . . . 8 |- ( ph -> w = w )
35	32, 33, 34	breq123d 4461	. . . . . . 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 2629	. . . . . . . . 9 |- F/_ x z
38		nfv 1683	. . . . . . . . . 10 |- F/ x ph
39		nffvmpt1 5872	. . . . . . . . . . . 12 |- F/_ x ( ( x e. A |-> R ) ` z )
40		nfcv 2629	. . . . . . . . . . . 12 |- F/_ x ( y e. B |-> S )
41		nfcv 2629	. . . . . . . . . . . 12 |- F/_ x w
42	39, 40, 41	nfbr 4491	. . . . . . . . . . 11 |- F/ x ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w
43		nfcsb1v 3451	. . . . . . . . . . . 12 |- F/_ x [_ z / x ]_ T
44	43	nfeq2 2646	. . . . . . . . . . 11 |- F/ x w = [_ z / x ]_ T
45	42, 44	nfbi 1881	. . . . . . . . . 10 |- F/ x ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T )
46	38, 45	nfim 1867	. . . . . . . . 9 |- F/ x ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
47		fveq2 5864	. . . . . . . . . . . 12 |- ( x = z -> ( ( x e. A |-> R ) ` x ) = ( ( x e. A |-> R ) ` z ) )
48	47	breq1d 4457	. . . . . . . . . . 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 3444	. . . . . . . . . . . 12 |- ( x = z -> T = [_ z / x ]_ T )
50	49	eqeq2d 2481	. . . . . . . . . . 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 3116	. . . . . . . . . . . 12 |- w e. _V
54		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> y = R )
55	54	eleq1d 2536	. . . . . . . . . . . . . 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 2489	. . . . . . . . . . . . . 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 4507	. . . . . . . . . . . . 13 |- ( y e. B |-> S ) = { <. y , u >. | ( y e. B /\ u = S ) }
62	60, 61	brabga 4761	. . . . . . . . . . . 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 5955	. . . . . . . . . . . . 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 4457	. . . . . . . . . . 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 3176	. . . . . . . 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 3116	. . . 4 |- z e. _V
78	77, 53	opelco 5172	. . 3 |- ( <. z , w >. e. ( G o. F ) <-> E. u ( z F u /\ u G w ) )
79		df-mpt 4507	. . . . 5 |- ( x e. A |-> T ) = { <. x , v >. | ( x e. A /\ v = T ) }
80	79	eleq2i 2545	. . . 4 |- ( <. z , w >. e. ( x e. A |-> T ) <-> <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } )
81		nfv 1683	. . . . . 6 |- F/ x z e. A
82	43	nfeq2 2646	. . . . . 6 |- F/ x v = [_ z / x ]_ T
83	81, 82	nfan 1875	. . . . 5 |- F/ x ( z e. A /\ v = [_ z / x ]_ T )
84		nfv 1683	. . . . 5 |- F/ v ( z e. A /\ w = [_ z / x ]_ T )
85		eleq1 2539	. . . . . 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 710	. . . . 5 |- ( x = z -> ( ( x e. A /\ v = T ) <-> ( z e. A /\ v = [_ z / x ]_ T ) ) )
88		eqeq1 2471	. . . . . 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 4772	. . . 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 5097	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   _Vcvv 3113   [_csb 3435   <.cop 4033   class class class wbr 4447   {copab 4504   |-> cmpt 4505   dom cdm 4999   o. ccom 5003   Rel wrel 5004   Fun wfun 5580   -->wf 5582   ` cfv 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594

This theorem is referenced by:  fmptcof  6053  fcompt  6055  fcoconst  6056  ofco  6542  ccatco  12758  lo1o12  13312  rlimcn1  13367  rlimcn1b  13368  rlimdiv  13424  ackbijnn  13596  setcepi  15266  prf1st  15324  prf2nd  15325  hofcllem  15378  prdsidlem  15763  pws0g  15767  pwsco1mhm  15808  pwsco2mhm  15809  pwsinvg  15979  pwssub  15980  galactghm  16220  efginvrel1  16539  frgpup3lem  16588  gsumzf1o  16705  gsumzf1oOLD  16708  gsumconst  16742  gsumzmhm  16745  gsumzmhmOLD  16746  gsummhm2  16749  gsummhm2OLD  16750  gsummptmhm  16751  gsumsub  16762  gsumsubOLD  16763  gsum2dlem2  16786  gsum2dOLD  16788  dprdfsub  16848  dprdfsubOLD  16855  lmhmvsca  17471  psrass1lem  17797  psrlinv  17818  psrcom  17832  evlslem2  17948  coe1fval3  18015  psropprmul  18047  coe1z  18072  coe1mul2  18078  coe1tm  18082  ply1coe  18105  ply1coeOLD  18106  evls1sca  18128  frgpcyg  18376  evpmodpmf1o  18396  mhmvlin  18663  ofco2  18717  mdetleib2  18854  mdetralt  18874  smadiadetlem3  18934  ptrescn  19872  lmcn2  19882  qtopeu  19949  flfcnp2  20240  tgpconcomp  20343  tsmsmhm  20380  tsmssub  20383  tsmsxplem1  20387  negfcncf  21155  pcopt  21254  pcopt2  21255  pi1xfrcnvlem  21288  ovolctb  21633  ovolfs2  21712  uniioombllem2  21724  uniioombllem3  21726  ismbf  21769  mbfconst  21774  ismbfcn2  21778  itg1climres  21853  iblabslem  21966  iblabs  21967  bddmulibl  21977  limccnp  22027  limccnp2  22028  limcco  22029  dvcof  22083  dvcjbr  22084  dvcj  22085  dvfre  22086  dvmptcj  22103  dvmptco  22107  dvcnvlem  22109  dvef  22113  dvlip  22126  dvlipcn  22127  itgsubstlem  22181  plypf1  22341  plyco  22370  dgrcolem1  22401  dgrcolem2  22402  dgrco  22403  plycjlem  22404  taylply2  22494  logcn  22753  leibpi  22998  efrlim  23024  jensenlem2  23042  amgmlem  23044  ftalem7  23077  lgseisenlem4  23352  dchrisum0  23430  cofmpt  27173  ofcfval4  27741  eulerpartgbij  27948  dstfrvclim1  28053  lgamgulmlem2  28209  lgamcvg2  28234  cvmliftlem6  28372  cvmliftphtlem  28399  cvmlift3lem5  28405  circum  28512  mblfinlem2  29627  volsupnfl  29634  itgaddnc  29650  itgmulc2nc  29658  ftc1anclem1  29665  ftc1anclem2  29666  ftc1anclem3  29667  ftc1anclem4  29668  ftc1anclem5  29669  ftc1anclem6  29670  ftc1anclem7  29671  ftc1anclem8  29672  fnopabco  29814  upixp  29821  mendassa  30748  dvcosax  31256  dirkercncflem4  31406  fourierdlem23  31430  fourierdlem101  31508  fourierdlem111  31518