Theorem caofinvl 5753

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 270	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> N : S --> S )
4		caofref.2	. . . . . . . . 9 |- ( ph -> F : A --> S )
5	4	ffvelrnda 5323	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelrnd 5324	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2081	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 5343	. . . . . 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 5054	. . . . . 6 |- ( ph -> ( G : A --> S <-> ( v e. A |-> ( N ` ( F ` v ) ) ) : A --> S ) )
11	8, 10	mpbird 165	. . . . 5 |- ( ph -> G : A --> S )
12	11	ffvelrnda 5323	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelrnda 5323	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14	6	ralrimiva 2434	. . . . . . 7 |- ( ph -> A. v e. A ( N ` ( F ` v ) ) e. S )
15	7	fnmpt 5045	. . . . . . 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 5009	. . . . . 6 |- ( ph -> ( G Fn A <-> ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A ) )
18	16, 17	mpbird 165	. . . . 5 |- ( ph -> G Fn A )
19		dffn5im 5240	. . . . 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 5247	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
22	1, 12, 13, 20, 21	offval2 5746	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
23	9	fveq1d 5200	. . . . . . . 8 |- ( ph -> ( G ` w ) = ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) )
24	23	adantr 270	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) )
25		simpr 108	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> w e. A )
26	2	adantr 270	. . . . . . . . 9 |- ( ( ph /\ w e. A ) -> N : S --> S )
27	26, 13	ffvelrnd 5324	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( N ` ( F ` w ) ) e. S )
28		fveq2 5198	. . . . . . . . . 10 |- ( v = w -> ( F ` v ) = ( F ` w ) )
29	28	fveq2d 5202	. . . . . . . . 9 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
30	29, 7	fvmptg 5269	. . . . . . . 8 |- ( ( w e. A /\ ( N ` ( F ` w ) ) e. S ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
31	25, 27, 30	syl2anc 403	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
32	24, 31	eqtrd 2113	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
33	32	oveq1d 5547	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
34		caofinvl.6	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )
35	34	ralrimiva 2434	. . . . . . 7 |- ( ph -> A. x e. S ( ( N ` x ) R x ) = B )
36	35	adantr 270	. . . . . 6 |- ( ( ph /\ w e. A ) -> A. x e. S ( ( N ` x ) R x ) = B )
37		fveq2 5198	. . . . . . . . 9 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
38		id 19	. . . . . . . . 9 |- ( x = ( F ` w ) -> x = ( F ` w ) )
39	37, 38	oveq12d 5550	. . . . . . . 8 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
40	39	eqeq1d 2089	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( ( N ` x ) R x ) = B <-> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B ) )
41	40	rspcva 2699	. . . . . 6 |- ( ( ( F ` w ) e. S /\ A. x e. S ( ( N ` x ) R x ) = B ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
42	13, 36, 41	syl2anc 403	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
43	33, 42	eqtrd 2113	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
44	43	mpteq2dva 3868	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
45	22, 44	eqtrd 2113	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
46		fconstmpt 4405	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
47	45, 46	syl6eqr 2131	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   {csn 3398   |-> cmpt 3839   X. cxp 4361   Fn wfn 4917   -->wf 4918   ` cfv 4922  (class class class)co 5532   oFcof 5730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-of 5732

This theorem is referenced by: (None)