Theorem fmptcof2 29457

Description: Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypotheses

Ref	Expression
fmptcof2.x	|- F/_ x S
fmptcof2.y	|- F/_ y T
fmptcof2.1	|- F/_ x A
fmptcof2.2	|- F/_ x B
fmptcof2.3	|- F/ x ph
fmptcof2.4	|- ( ph -> A. x e. A R e. B )
fmptcof2.5	|- ( ph -> F = ( x e. A |-> R ) )
fmptcof2.6	|- ( ph -> G = ( y e. B |-> S ) )
fmptcof2.7	|- ( y = R -> S = T )

Assertion

Ref	Expression
fmptcof2	|- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Distinct variable groups:   x, y   y, B   y, R

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

Proof of Theorem fmptcof2

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

Step	Hyp	Ref	Expression
1		relco 5633	. 2 |- Rel ( G o. F )
2		funmpt 5926	. . 3 |- Fun ( x e. A |-> T )
3		funrel 5905	. . 3 |- ( Fun ( x e. A |-> T ) -> Rel ( x e. A |-> T ) )
4	2, 3	ax-mp 5	. 2 |- Rel ( x e. A |-> T )
5		fmptcof2.3	. . . . . . . . . . . . 13 |- F/ x ph
6		fmptcof2.1	. . . . . . . . . . . . 13 |- F/_ x A
7		fmptcof2.2	. . . . . . . . . . . . 13 |- F/_ x B
8		fmptcof2.4	. . . . . . . . . . . . . 14 |- ( ph -> A. x e. A R e. B )
9	8	r19.21bi 2932	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> R e. B )
10		eqid 2622	. . . . . . . . . . . . 13 |- ( x e. A |-> R ) = ( x e. A |-> R )
11	5, 6, 7, 9, 10	fmptdF 29456	. . . . . . . . . . . 12 |- ( ph -> ( x e. A |-> R ) : A --> B )
12		fmptcof2.5	. . . . . . . . . . . . 13 |- ( ph -> F = ( x e. A |-> R ) )
13	12	feq1d 6030	. . . . . . . . . . . 12 |- ( ph -> ( F : A --> B <-> ( x e. A |-> R ) : A --> B ) )
14	11, 13	mpbird 247	. . . . . . . . . . 11 |- ( ph -> F : A --> B )
15		ffun 6048	. . . . . . . . . . 11 |- ( F : A --> B -> Fun F )
16	14, 15	syl 17	. . . . . . . . . 10 |- ( ph -> Fun F )
17		funbrfv 6234	. . . . . . . . . . 11 |- ( Fun F -> ( z F u -> ( F ` z ) = u ) )
18	17	imp 445	. . . . . . . . . 10 |- ( ( Fun F /\ z F u ) -> ( F ` z ) = u )
19	16, 18	sylan 488	. . . . . . . . 9 |- ( ( ph /\ z F u ) -> ( F ` z ) = u )
20	19	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ z F u ) -> u = ( F ` z ) )
21	20	a1d 25	. . . . . . 7 |- ( ( ph /\ z F u ) -> ( u G w -> u = ( F ` z ) ) )
22	21	expimpd 629	. . . . . 6 |- ( ph -> ( ( z F u /\ u G w ) -> u = ( F ` z ) ) )
23	22	pm4.71rd 667	. . . . 5 |- ( ph -> ( ( z F u /\ u G w ) <-> ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
24	23	exbidv 1850	. . . 4 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) ) )
25		fvex 6201	. . . . . 6 |- ( F ` z ) e. _V
26		breq2 4657	. . . . . . 7 |- ( u = ( F ` z ) -> ( z F u <-> z F ( F ` z ) ) )
27		breq1 4656	. . . . . . 7 |- ( u = ( F ` z ) -> ( u G w <-> ( F ` z ) G w ) )
28	26, 27	anbi12d 747	. . . . . 6 |- ( u = ( F ` z ) -> ( ( z F u /\ u G w ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) ) )
29	25, 28	ceqsexv 3242	. . . . 5 |- ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z F ( F ` z ) /\ ( F ` z ) G w ) )
30		funfvbrb 6330	. . . . . . . . 9 |- ( Fun F -> ( z e. dom F <-> z F ( F ` z ) ) )
31	16, 30	syl 17	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z F ( F ` z ) ) )
32		fdm 6051	. . . . . . . . . 10 |- ( F : A --> B -> dom F = A )
33	14, 32	syl 17	. . . . . . . . 9 |- ( ph -> dom F = A )
34	33	eleq2d 2687	. . . . . . . 8 |- ( ph -> ( z e. dom F <-> z e. A ) )
35	31, 34	bitr3d 270	. . . . . . 7 |- ( ph -> ( z F ( F ` z ) <-> z e. A ) )
36	12	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( F ` z ) = ( ( x e. A |-> R ) ` z ) )
37		fmptcof2.6	. . . . . . . 8 |- ( ph -> G = ( y e. B |-> S ) )
38		eqidd 2623	. . . . . . . 8 |- ( ph -> w = w )
39	36, 37, 38	breq123d 4667	. . . . . . 7 |- ( ph -> ( ( F ` z ) G w <-> ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) )
40	35, 39	anbi12d 747	. . . . . 6 |- ( ph -> ( ( z F ( F ` z ) /\ ( F ` z ) G w ) <-> ( z e. A /\ ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) ) )
41	6	nfcri 2758	. . . . . . . . . 10 |- F/ x z e. A
42		nffvmpt1 6199	. . . . . . . . . . . . 13 |- F/_ x ( ( x e. A |-> R ) ` z )
43		fmptcof2.x	. . . . . . . . . . . . . 14 |- F/_ x S
44	7, 43	nfmpt 4746	. . . . . . . . . . . . 13 |- F/_ x ( y e. B |-> S )
45		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x w
46	42, 44, 45	nfbr 4699	. . . . . . . . . . . 12 |- F/ x ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w
47		nfcsb1v 3549	. . . . . . . . . . . . 13 |- F/_ x [_ z / x ]_ T
48	47	nfeq2 2780	. . . . . . . . . . . 12 |- F/ x w = [_ z / x ]_ T
49	46, 48	nfbi 1833	. . . . . . . . . . 11 |- F/ x ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T )
50	5, 49	nfim 1825	. . . . . . . . . 10 |- F/ x ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
51	41, 50	nfim 1825	. . . . . . . . 9 |- F/ x ( z e. A -> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) )
52		eleq1 2689	. . . . . . . . . 10 |- ( x = z -> ( x e. A <-> z e. A ) )
53		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = z -> ( ( x e. A |-> R ) ` x ) = ( ( x e. A |-> R ) ` z ) )
54	53	breq1d 4663	. . . . . . . . . . . 12 |- ( x = z -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) )
55		csbeq1a 3542	. . . . . . . . . . . . 13 |- ( x = z -> T = [_ z / x ]_ T )
56	55	eqeq2d 2632	. . . . . . . . . . . 12 |- ( x = z -> ( w = T <-> w = [_ z / x ]_ T ) )
57	54, 56	bibi12d 335	. . . . . . . . . . 11 |- ( 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 ) ) )
58	57	imbi2d 330	. . . . . . . . . 10 |- ( 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 ) ) ) )
59	52, 58	imbi12d 334	. . . . . . . . 9 |- ( x = z -> ( ( x e. A -> ( ph -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) ) ) <-> ( z e. A -> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) ) ) )
60		vex 3203	. . . . . . . . . . . 12 |- w e. _V
61		nfv 1843	. . . . . . . . . . . . . 14 |- F/ y R e. B
62		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ y w
63		fmptcof2.y	. . . . . . . . . . . . . . 15 |- F/_ y T
64	62, 63	nfeq 2776	. . . . . . . . . . . . . 14 |- F/ y w = T
65	61, 64	nfan 1828	. . . . . . . . . . . . 13 |- F/ y ( R e. B /\ w = T )
66		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> y = R )
67	66	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( y e. B <-> R e. B ) )
68		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> u = w )
69		fmptcof2.7	. . . . . . . . . . . . . . . 16 |- ( y = R -> S = T )
70	69	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( y = R /\ u = w ) -> S = T )
71	68, 70	eqeq12d 2637	. . . . . . . . . . . . . 14 |- ( ( y = R /\ u = w ) -> ( u = S <-> w = T ) )
72	67, 71	anbi12d 747	. . . . . . . . . . . . 13 |- ( ( y = R /\ u = w ) -> ( ( y e. B /\ u = S ) <-> ( R e. B /\ w = T ) ) )
73		df-mpt 4730	. . . . . . . . . . . . 13 |- ( y e. B |-> S ) = { <. y , u >. | ( y e. B /\ u = S ) }
74	65, 72, 73	brabgaf 29420	. . . . . . . . . . . 12 |- ( ( R e. B /\ w e. _V ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
75	9, 60, 74	sylancl 694	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( R ( y e. B |-> S ) w <-> ( R e. B /\ w = T ) ) )
76		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> x e. A )
77	6	fvmpt2f 6283	. . . . . . . . . . . . 13 |- ( ( x e. A /\ R e. B ) -> ( ( x e. A |-> R ) ` x ) = R )
78	76, 9, 77	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> R ) ` x ) = R )
79	78	breq1d 4663	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> R ( y e. B |-> S ) w ) )
80	9	biantrurd 529	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( w = T <-> ( R e. B /\ w = T ) ) )
81	75, 79, 80	3bitr4d 300	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) )
82	81	expcom 451	. . . . . . . . 9 |- ( x e. A -> ( ph -> ( ( ( x e. A |-> R ) ` x ) ( y e. B |-> S ) w <-> w = T ) ) )
83	51, 59, 82	chvar 2262	. . . . . . . 8 |- ( z e. A -> ( ph -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) ) )
84	83	impcom 446	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w <-> w = [_ z / x ]_ T ) )
85	84	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( z e. A /\ ( ( x e. A |-> R ) ` z ) ( y e. B |-> S ) w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
86	40, 85	bitrd 268	. . . . 5 |- ( ph -> ( ( z F ( F ` z ) /\ ( F ` z ) G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
87	29, 86	syl5bb 272	. . . 4 |- ( ph -> ( E. u ( u = ( F ` z ) /\ ( z F u /\ u G w ) ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
88	24, 87	bitrd 268	. . 3 |- ( ph -> ( E. u ( z F u /\ u G w ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
89		vex 3203	. . . 4 |- z e. _V
90	89, 60	opelco 5293	. . 3 |- ( <. z , w >. e. ( G o. F ) <-> E. u ( z F u /\ u G w ) )
91		df-mpt 4730	. . . . 5 |- ( x e. A |-> T ) = { <. x , v >. | ( x e. A /\ v = T ) }
92	91	eleq2i 2693	. . . 4 |- ( <. z , w >. e. ( x e. A |-> T ) <-> <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } )
93	47	nfeq2 2780	. . . . . 6 |- F/ x v = [_ z / x ]_ T
94	41, 93	nfan 1828	. . . . 5 |- F/ x ( z e. A /\ v = [_ z / x ]_ T )
95		nfv 1843	. . . . 5 |- F/ v ( z e. A /\ w = [_ z / x ]_ T )
96	55	eqeq2d 2632	. . . . . 6 |- ( x = z -> ( v = T <-> v = [_ z / x ]_ T ) )
97	52, 96	anbi12d 747	. . . . 5 |- ( x = z -> ( ( x e. A /\ v = T ) <-> ( z e. A /\ v = [_ z / x ]_ T ) ) )
98		eqeq1 2626	. . . . . 6 |- ( v = w -> ( v = [_ z / x ]_ T <-> w = [_ z / x ]_ T ) )
99	98	anbi2d 740	. . . . 5 |- ( v = w -> ( ( z e. A /\ v = [_ z / x ]_ T ) <-> ( z e. A /\ w = [_ z / x ]_ T ) ) )
100	94, 95, 89, 60, 97, 99	opelopabf 5000	. . . 4 |- ( <. z , w >. e. { <. x , v >. | ( x e. A /\ v = T ) } <-> ( z e. A /\ w = [_ z / x ]_ T ) )
101	92, 100	bitri 264	. . 3 |- ( <. z , w >. e. ( x e. A |-> T ) <-> ( z e. A /\ w = [_ z / x ]_ T ) )
102	88, 90, 101	3bitr4g 303	. 2 |- ( ph -> ( <. z , w >. e. ( G o. F ) <-> <. z , w >. e. ( x e. A |-> T ) ) )
103	1, 4, 102	eqrelrdv 5216	1 |- ( ph -> ( G o. F ) = ( x e. A |-> T ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   [_csb 3533   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   -->wf 5884   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-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-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-fv 5896

This theorem is referenced by:  esumf1o  30112