Theorem caofinvl 6318

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 5834	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelcdmd 5835	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2238	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 5853	. . . . . 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 5515	. . . . . 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 5834	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelcdmda 5834	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14	6	ralrimiva 2623	. . . . . . 7 |- ( ph -> A. v e. A ( N ` ( F ` v ) ) e. S )
15	7	fnmpt 5505	. . . . . . 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 5466	. . . . . 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 5742	. . . . 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 5750	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
22	1, 12, 13, 20, 21	offval2 6308	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
23	9	fveq1d 5692	. . . . . . . 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 5835	. . . . . . . 8 |- ( ( ph /\ w e. A ) -> ( N ` ( F ` w ) ) e. S )
28		fveq2 5690	. . . . . . . . . 10 |- ( v = w -> ( F ` v ) = ( F ` w ) )
29	28	fveq2d 5694	. . . . . . . . 9 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
30	29, 7	fvmptg 5775	. . . . . . . 8 |- ( ( w e. A /\ ( N ` ( F ` w ) ) e. S ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
31	25, 27, 30	syl2anc 415	. . . . . . 7 |- ( ( ph /\ w e. A ) -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
32	24, 31	eqtrd 2271	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
33	32	oveq1d 6090	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
34		fveq2 5690	. . . . . . . 8 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
35		id 19	. . . . . . . 8 |- ( x = ( F ` w ) -> x = ( F ` w ) )
36	34, 35	oveq12d 6093	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
37	36	eqeq1d 2247	. . . . . 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 2623	. . . . . . 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 2934	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
42	33, 41	eqtrd 2271	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
43	42	mpteq2dva 4216	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
44	22, 43	eqtrd 2271	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
45		fconstmpt 4817	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
46	44, 45	eqtr4di 2289	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3705   |-> cmpt 4187   X. cxp 4767   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   oFcof 6290

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292

This theorem is referenced by: (None)