Theorem caofid0l 6235

Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofid0.3	|- ( ph -> B e. W )
caofid0l.5	|- ( ( ph /\ x e. S ) -> ( B R x ) = x )

Assertion

Ref	Expression
caofid0l	|- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Distinct variable groups:   x, B   x, F   ph, x   x, R   x, S

Allowed substitution hints:   A( x)   V( x)   W( x)

Proof of Theorem caofid0l

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 |- ( ph -> A e. V )
2		caofid0.3	. . 3 |- ( ph -> B e. W )
3		fnconstg 5519	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
4	2, 3	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
5		caofref.2	. . 3 |- ( ph -> F : A --> S )
6	5	ffnd 5470	. 2 |- ( ph -> F Fn A )
7		fvconst2g 5846	. . 3 |- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
8	2, 7	sylan 283	. 2 |- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
9		eqidd 2230	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
10		caofid0l.5	. . . 4 |- ( ( ph /\ x e. S ) -> ( B R x ) = x )
11	10	ralrimiva 2603	. . 3 |- ( ph -> A. x e. S ( B R x ) = x )
12	5	ffvelcdmda 5763	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
13		oveq2 6002	. . . . 5 |- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )
14		id 19	. . . . 5 |- ( x = ( F ` w ) -> x = ( F ` w ) )
15	13, 14	eqeq12d 2244	. . . 4 |- ( x = ( F ` w ) -> ( ( B R x ) = x <-> ( B R ( F ` w ) ) = ( F ` w ) ) )
16	15	rspccva 2906	. . 3 |- ( ( A. x e. S ( B R x ) = x /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = ( F ` w ) )
17	11, 12, 16	syl2an2r 597	. 2 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( F ` w ) )
18	1, 4, 6, 6, 8, 9, 17	offveq 6229	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   {csn 3666   X. cxp 4714   Fn wfn 5309   -->wf 5310   ` cfv 5314  (class class class)co 5994   oFcof 6206

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4626

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-of 6208

This theorem is referenced by:  psr0lid  14631