Theorem caofinvl 6107

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 5653	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelcdmd 5654	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2177	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 5672	. . . . . 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 5354	. . . . . 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 5653	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelcdmda 5653	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14	6	ralrimiva 2550	. . . . . . 7 |- ( ph -> A. v e. A ( N ` ( F ` v ) ) e. S )
15	7	fnmpt 5344	. . . . . . 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 5308	. . . . . 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 5563	. . . . 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 5571	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
22	1, 12, 13, 20, 21	offval2 6100	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
23	9	fveq1d 5519	. . . . . . . 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 5654	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( N ` ( F ` w ) ) e. S )
28		fveq2 5517	. . . . . . . . . 10 |- ( v = w -> ( F ` v ) = ( F ` w ) )
29	28	fveq2d 5521	. . . . . . . . 9 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
30	29, 7	fvmptg 5594	. . . . . . . 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 2210	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
33	32	oveq1d 5892	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
34		fveq2 5517	. . . . . . . 8 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
35		id 19	. . . . . . . 8 |- ( x = ( F ` w ) -> x = ( F ` w ) )
36	34, 35	oveq12d 5895	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
37	36	eqeq1d 2186	. . . . . 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 2550	. . . . . . 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 2848	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
42	33, 41	eqtrd 2210	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
43	42	mpteq2dva 4095	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
44	22, 43	eqtrd 2210	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
45		fconstmpt 4675	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
46	44, 45	eqtr4di 2228	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   {csn 3594   |-> cmpt 4066   X. cxp 4626   Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5877   oFcof 6083

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085

This theorem is referenced by: (None)