Theorem caofinvl 6183

Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofinv.3	|- ( ph -> B e. W )
caofinv.4	|- ( ph -> N : S --> S )
caofinv.5	|- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) )
caofinvl.6	|- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )

Assertion

Ref	Expression
caofinvl	|- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Distinct variable groups:   x, B   x, F   x, G   ph, x   x, R   x, S   v, A   v, F, x   x, N, v   v, S   ph, v

Allowed substitution hints:   A( x)   B( v)   R( v)   G( v)   V( x, v)   W( x, v)

Proof of Theorem caofinvl

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. . . 4 |- ( ph -> A e. V )
2		caofinv.4	. . . . . . . . 9 |- ( ph -> N : S --> S )
3	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> N : S --> S )
4		caofref.2	. . . . . . . . 9 |- ( ph -> F : A --> S )
5	4	ffvelcdmda 5714	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelcdmd 5715	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2204	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 5733	. . . . . 6 |- ( ph -> ( v e. A |-> ( N ` ( F ` v ) ) ) : A --> S )
9		caofinv.5	. . . . . . 7 |- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) )
10	9	feq1d 5411	. . . . . 6 |- ( ph -> ( G : A --> S <-> ( v e. A |-> ( N ` ( F ` v ) ) ) : A --> S ) )
11	8, 10	mpbird 167	. . . . 5 |- ( ph -> G : A --> S )
12	11	ffvelcdmda 5714	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelcdmda 5714	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14	6	ralrimiva 2578	. . . . . . 7 |- ( ph -> A. v e. A ( N ` ( F ` v ) ) e. S )
15	7	fnmpt 5401	. . . . . . 7 |- ( A. v e. A ( N ` ( F ` v ) ) e. S -> ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A )
16	14, 15	syl 14	. . . . . 6 |- ( ph -> ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A )
17	9	fneq1d 5363	. . . . . 6 |- ( ph -> ( G Fn A <-> ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A ) )
18	16, 17	mpbird 167	. . . . 5 |- ( ph -> G Fn A )
19		dffn5im 5623	. . . . 5 |- ( G Fn A -> G = ( w e. A |-> ( G ` w ) ) )
20	18, 19	syl 14	. . . 4 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
21	4	feqmptd 5631	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
22	1, 12, 13, 20, 21	offval2 6173	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
23	9	fveq1d 5577	. . . . . . . 8 |- ( ph -> ( G ` w ) = ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) )
24	23	adantr 276	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) )
25		simpr 110	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> w e. A )
26	2	adantr 276	. . . . . . . . 9 |- ( ( ph /\ w e. A ) -> N : S --> S )
27	26, 13	ffvelcdmd 5715	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( N ` ( F ` w ) ) e. S )
28		fveq2 5575	. . . . . . . . . 10 |- ( v = w -> ( F ` v ) = ( F ` w ) )
29	28	fveq2d 5579	. . . . . . . . 9 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
30	29, 7	fvmptg 5654	. . . . . . . 8 |- ( ( w e. A /\ ( N ` ( F ` w ) ) e. S ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
31	25, 27, 30	syl2anc 411	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
32	24, 31	eqtrd 2237	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
33	32	oveq1d 5958	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
34		fveq2 5575	. . . . . . . 8 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
35		id 19	. . . . . . . 8 |- ( x = ( F ` w ) -> x = ( F ` w ) )
36	34, 35	oveq12d 5961	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
37	36	eqeq1d 2213	. . . . . 6 |- ( x = ( F ` w ) -> ( ( ( N ` x ) R x ) = B <-> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B ) )
38		caofinvl.6	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )
39	38	ralrimiva 2578	. . . . . . 7 |- ( ph -> A. x e. S ( ( N ` x ) R x ) = B )
40	39	adantr 276	. . . . . 6 |- ( ( ph /\ w e. A ) -> A. x e. S ( ( N ` x ) R x ) = B )
41	37, 40, 13	rspcdva 2881	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
42	33, 41	eqtrd 2237	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
43	42	mpteq2dva 4133	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
44	22, 43	eqtrd 2237	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
45		fconstmpt 4721	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
46	44, 45	eqtr4di 2255	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   A.wral 2483   {csn 3632   |-> cmpt 4104   X. cxp 4672   Fn wfn 5265   -->wf 5266   ` cfv 5270  (class class class)co 5943   oFcof 6155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-of 6157

This theorem is referenced by: (None)