Theorem caofinvl 6924

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 481	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> N : S --> S )
4		caofref.2	. . . . . . . . 9 |- ( ph -> F : A --> S )
5	4	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. S )
6	3, 5	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ v e. A ) -> ( N ` ( F ` v ) ) e. S )
7		eqid 2622	. . . . . . 7 |- ( v e. A |-> ( N ` ( F ` v ) ) ) = ( v e. A |-> ( N ` ( F ` v ) ) )
8	6, 7	fmptd 6385	. . . . . 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 6030	. . . . . 6 |- ( ph -> ( G : A --> S <-> ( v e. A |-> ( N ` ( F ` v ) ) ) : A --> S ) )
11	8, 10	mpbird 247	. . . . 5 |- ( ph -> G : A --> S )
12	11	ffvelrnda 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
13	4	ffvelrnda 6359	. . . 4 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
14		fvex 6201	. . . . . . 7 |- ( N ` ( F ` v ) ) e. _V
15	14, 7	fnmpti 6022	. . . . . 6 |- ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A
16	9	fneq1d 5981	. . . . . 6 |- ( ph -> ( G Fn A <-> ( v e. A |-> ( N ` ( F ` v ) ) ) Fn A ) )
17	15, 16	mpbiri 248	. . . . 5 |- ( ph -> G Fn A )
18		dffn5 6241	. . . . 5 |- ( G Fn A <-> G = ( w e. A |-> ( G ` w ) ) )
19	17, 18	sylib 208	. . . 4 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
20	4	feqmptd 6249	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
21	1, 12, 13, 19, 20	offval2 6914	. . 3 |- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) )
22	9	fveq1d 6193	. . . . . . 7 |- ( ph -> ( G ` w ) = ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) )
23		fveq2 6191	. . . . . . . . 9 |- ( v = w -> ( F ` v ) = ( F ` w ) )
24	23	fveq2d 6195	. . . . . . . 8 |- ( v = w -> ( N ` ( F ` v ) ) = ( N ` ( F ` w ) ) )
25		fvex 6201	. . . . . . . 8 |- ( N ` ( F ` w ) ) e. _V
26	24, 7, 25	fvmpt 6282	. . . . . . 7 |- ( w e. A -> ( ( v e. A |-> ( N ` ( F ` v ) ) ) ` w ) = ( N ` ( F ` w ) ) )
27	22, 26	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ w e. A ) -> ( G ` w ) = ( N ` ( F ` w ) ) )
28	27	oveq1d 6665	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
29		caofinvl.6	. . . . . . . 8 |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )
30	29	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. S ( ( N ` x ) R x ) = B )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ w e. A ) -> A. x e. S ( ( N ` x ) R x ) = B )
32		fveq2 6191	. . . . . . . . 9 |- ( x = ( F ` w ) -> ( N ` x ) = ( N ` ( F ` w ) ) )
33		id 22	. . . . . . . . 9 |- ( x = ( F ` w ) -> x = ( F ` w ) )
34	32, 33	oveq12d 6668	. . . . . . . 8 |- ( x = ( F ` w ) -> ( ( N ` x ) R x ) = ( ( N ` ( F ` w ) ) R ( F ` w ) ) )
35	34	eqeq1d 2624	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( ( N ` x ) R x ) = B <-> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B ) )
36	35	rspcva 3307	. . . . . 6 |- ( ( ( F ` w ) e. S /\ A. x e. S ( ( N ` x ) R x ) = B ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
37	13, 31, 36	syl2anc 693	. . . . 5 |- ( ( ph /\ w e. A ) -> ( ( N ` ( F ` w ) ) R ( F ` w ) ) = B )
38	28, 37	eqtrd 2656	. . . 4 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) R ( F ` w ) ) = B )
39	38	mpteq2dva 4744	. . 3 |- ( ph -> ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) = ( w e. A |-> B ) )
40	21, 39	eqtrd 2656	. 2 |- ( ph -> ( G oF R F ) = ( w e. A |-> B ) )
41		fconstmpt 5163	. 2 |- ( A X. { B } ) = ( w e. A |-> B )
42	40, 41	syl6eqr 2674	1 |- ( ph -> ( G oF R F ) = ( A X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  grpvlinv  20201  lflnegl  34363