Theorem caofinvl 6155

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 5693	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelcdmd 5694	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2193	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 5712	. . . . . 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 5390	. . . . . 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 5693	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelcdmda 5693	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14	6	ralrimiva 2567	. . . . . . 7 |- ( ph -> A. v e. A ( N ` ( F ` v ) ) e. S )
15	7	fnmpt 5380	. . . . . . 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 5344	. . . . . 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 5602	. . . . 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 5610	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
22	1, 12, 13, 20, 21	offval2 6146	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
23	9	fveq1d 5556	. . . . . . . 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 5694	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( N ` ( F ` w ) ) e. S )
28		fveq2 5554	. . . . . . . . . 10 |- ( v = w -> ( F ` v ) = ( F ` w ) )
29	28	fveq2d 5558	. . . . . . . . 9 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
30	29, 7	fvmptg 5633	. . . . . . . 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 2226	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
33	32	oveq1d 5933	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
34		fveq2 5554	. . . . . . . 8 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
35		id 19	. . . . . . . 8 |- ( x = ( F ` w ) -> x = ( F ` w ) )
36	34, 35	oveq12d 5936	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
37	36	eqeq1d 2202	. . . . . 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 2567	. . . . . . 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 2869	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
42	33, 41	eqtrd 2226	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
43	42	mpteq2dva 4119	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
44	22, 43	eqtrd 2226	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
45		fconstmpt 4706	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
46	44, 45	eqtr4di 2244	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   {csn 3618   |-> cmpt 4090   X. cxp 4657   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   oFcof 6128

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by: (None)