Theorem fmptco 5986

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 5445	. 2 |- Rel ( G o. F )
2		funmpt 5563	. . 3 |- Fun ( x e. A |-> T )
3		funrel 5544	. . 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 2454	. . . . . . . . . . . . 13 |- ( x e. A |-> R ) = ( x e. A |-> R )
7	5, 6	fmptd 5977	. . . . . . . . . . . 12 |- ( ph -> ( x e. A |-> R ) : A --> B )
8		fmptco.2	. . . . . . . . . . . . 13 |- ( ph -> F = ( x e. A |-> R ) )
9	8	feq1d 5655	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> B <-> ( x e. A |-> R ) : A --> B ) )
10	7, 9	mpbird 232	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
11		ffun 5670	. . . . . . . . . . 11 |- ( F : A --> B -> Fun F )
12	10, 11	syl 16	. . . . . . . . . 10 |- ( ph -> Fun F )
13		funbrfv 5840	. . . . . . . . . . 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 2462	. . . . . . . 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 1681	. . . 4 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
21		fvex 5810	. . . . . 6 |- ( F ` z ) e. _V
22		breq2 4405	. . . . . . 7 |- ( u = ( F ` z ) -> ( z F u <-> z F ( F ` z ) ) )
23		breq1 4404	. . . . . . 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 3115	. . . . 5 |- ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) )
26		funfvbrb 5926	. . . . . . . . 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 5672	. . . . . . . . . 10 |- ( F : A --> B -> dom F = A )
29	10, 28	syl 16	. . . . . . . . 9 |- ( ph -> dom F = A )
30	29	eleq2d 2524	. . . . . . . 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 5802	. . . . . . . 8 |- ( ph -> ( F ` z ) = ( ( x e. A |-> R ) ` z ) )
33		fmptco.3	. . . . . . . 8 |- ( ph -> G = ( y e. B |-> S ) )
34		eqidd 2455	. . . . . . . 8 |- ( ph -> w = w )
35	32, 33, 34	breq123d 4415	. . . . . . 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 2616	. . . . . . . . 9 |- F/_ x z
38		nfv 1674	. . . . . . . . . 10 |- F/ x ph
39		nffvmpt1 5808	. . . . . . . . . . . 12 |- F/_ x ( ( x e. A |-> R ) ` z )
40		nfcv 2616	. . . . . . . . . . . 12 |- F/_ x ( y e. B |-> S )
41		nfcv 2616	. . . . . . . . . . . 12 |- F/_ x w
42	39, 40, 41	nfbr 4445	. . . . . . . . . . 11 |- F/ x ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w
43		nfcsb1v 3412	. . . . . . . . . . . 12 |- F/_ x [_ z / x ]_ T
44	43	nfeq2 2633	. . . . . . . . . . 11 |- F/ x w = [_ z / x ]_ T
45	42, 44	nfbi 1872	. . . . . . . . . 10 |- F/ x ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T )
46	38, 45	nfim 1858	. . . . . . . . 9 |- F/ x ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
47		fveq2 5800	. . . . . . . . . . . 12 |- ( x = z -> ( ( x e. A |-> R ) ` x ) = ( ( x e. A |-> R ) ` z ) )
48	47	breq1d 4411	. . . . . . . . . . 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 3405	. . . . . . . . . . . 12 |- ( x = z -> T = [_ z / x ]_ T )
50	49	eqeq2d 2468	. . . . . . . . . . 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 3081	. . . . . . . . . . . 12 |- w e. _V
54		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> y = R )
55	54	eleq1d 2523	. . . . . . . . . . . . . 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 2476	. . . . . . . . . . . . . 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 4461	. . . . . . . . . . . . 13 |- ( y e. B |-> S ) = { <. y , u >. | ( y e. B /\ u = S ) }
62	60, 61	brabga 4712	. . . . . . . . . . . 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 5891	. . . . . . . . . . . . 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 4411	. . . . . . . . . . 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 3141	. . . . . . . 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 3081	. . . 4 |- z e. _V
78	77, 53	opelco 5120	. . 3 |- ( <. z , w >. e. ( G o. F ) <-> E. u ( z F u /\ u G w ) )
79		df-mpt 4461	. . . . 5 |- ( x e. A |-> T ) = { <. x , v >. | ( x e. A /\ v = T ) }
80	79	eleq2i 2532	. . . 4 |- ( <. z , w >. e. ( x e. A |-> T ) <-> <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } )
81		nfv 1674	. . . . . 6 |- F/ x z e. A
82	43	nfeq2 2633	. . . . . 6 |- F/ x v = [_ z / x ]_ T
83	81, 82	nfan 1866	. . . . 5 |- F/ x ( z e. A /\ v = [_ z / x ]_ T )
84		nfv 1674	. . . . 5 |- F/ v ( z e. A /\ w = [_ z / x ]_ T )
85		eleq1 2526	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
86	49	eqeq2d 2468	. . . . . 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 2458	. . . . . 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 4722	. . . 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 5045	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   _Vcvv 3078   [_csb 3396   <.cop 3992   class class class wbr 4401   {copab 4458   |-> cmpt 4459   dom cdm 4949   o. ccom 4953   Rel wrel 4954   Fun wfun 5521   -->wf 5523   ` cfv 5527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535

This theorem is referenced by:  fmptcof  5987  fcompt  5989  fcoconst  5990  ofco  6451  ccatco  12582  lo1o12  13130  rlimcn1  13185  rlimcn1b  13186  rlimdiv  13242  ackbijnn  13410  setcepi  15076  prf1st  15134  prf2nd  15135  hofcllem  15188  prdsidlem  15573  pws0g  15577  pwsco1mhm  15618  pwsco2mhm  15619  pwsinvg  15787  pwssub  15788  galactghm  16028  efginvrel1  16347  frgpup3lem  16396  gsumzf1o  16513  gsumzf1oOLD  16516  gsumconst  16550  gsumzmhm  16553  gsumzmhmOLD  16554  gsummhm2  16557  gsummhm2OLD  16558  gsummptmhm  16559  gsumsub  16570  gsumsubOLD  16571  gsum2dlem2  16585  gsum2dOLD  16587  dprdfsub  16634  dprdfsubOLD  16641  lmhmvsca  17250  psrass1lem  17571  psrlinv  17592  psrcom  17606  evlslem2  17722  coe1fval3  17789  psropprmul  17817  coe1z  17841  coe1mul2  17847  coe1tm  17851  ply1coe  17872  ply1coeOLD  17873  evls1sca  17884  frgpcyg  18132  evpmodpmf1o  18152  mhmvlin  18423  ofco2  18460  mdetleib2  18527  mdetralt  18547  smadiadetlem3  18607  ptrescn  19345  lmcn2  19355  qtopeu  19422  flfcnp2  19713  tgpconcomp  19816  tsmsmhm  19853  tsmssub  19856  tsmsxplem1  19860  negfcncf  20628  pcopt  20727  pcopt2  20728  pi1xfrcnvlem  20761  ovolctb  21106  ovolfs2  21185  uniioombllem2  21197  uniioombllem3  21199  ismbf  21242  mbfconst  21247  ismbfcn2  21251  itg1climres  21326  iblabslem  21439  iblabs  21440  bddmulibl  21450  limccnp  21500  limccnp2  21501  limcco  21502  dvcof  21556  dvcjbr  21557  dvcj  21558  dvfre  21559  dvmptcj  21576  dvmptco  21580  dvcnvlem  21582  dvef  21586  dvlip  21599  dvlipcn  21600  itgsubstlem  21654  plypf1  21814  plyco  21843  dgrcolem1  21874  dgrcolem2  21875  dgrco  21876  plycjlem  21877  taylply2  21967  logcn  22226  leibpi  22471  efrlim  22497  jensenlem2  22515  amgmlem  22517  ftalem7  22550  lgseisenlem4  22825  dchrisum0  22903  cofmpt  26133  ofcfval4  26693  eulerpartgbij  26900  dstfrvclim1  27005  lgamgulmlem2  27161  lgamcvg2  27186  cvmliftlem6  27324  cvmliftphtlem  27351  cvmlift3lem5  27357  circum  27464  mblfinlem2  28578  volsupnfl  28585  itgaddnc  28601  itgmulc2nc  28609  ftc1anclem1  28616  ftc1anclem2  28617  ftc1anclem3  28618  ftc1anclem4  28619  ftc1anclem5  28620  ftc1anclem6  28621  ftc1anclem7  28622  ftc1anclem8  28623  fnopabco  28765  upixp  28772  mendassa  29700