Theorem offval3 6040

Description: General value of ( F oF R G ) with no assumptions on functionality of F and G. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
offval3	|- ( ( F e. V /\ G e. W ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )

Distinct variable groups:   x, F   x, G   x, V   x, W   x, R

Proof of Theorem offval3

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2700	. . 3 |- ( F e. V -> F e. _V )
2	1	adantr 274	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
3		elex 2700	. . 3 |- ( G e. W -> G e. _V )
4	3	adantl 275	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
5		dmexg 4811	. . . 4 |- ( F e. V -> dom F e. _V )
6		inex1g 4072	. . . 4 |- ( dom F e. _V -> ( dom F i^i dom G ) e. _V )
7		mptexg 5653	. . . 4 |- ( ( dom F i^i dom G ) e. _V -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
8	5, 6, 7	3syl 17	. . 3 |- ( F e. V -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
9	8	adantr 274	. 2 |- ( ( F e. V /\ G e. W ) -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
10		dmeq 4747	. . . . 5 |- ( a = F -> dom a = dom F )
11		dmeq 4747	. . . . 5 |- ( b = G -> dom b = dom G )
12	10, 11	ineqan12d 3284	. . . 4 |- ( ( a = F /\ b = G ) -> ( dom a i^i dom b ) = ( dom F i^i dom G ) )
13		fveq1 5428	. . . . 5 |- ( a = F -> ( a ` x ) = ( F ` x ) )
14		fveq1 5428	. . . . 5 |- ( b = G -> ( b ` x ) = ( G ` x ) )
15	13, 14	oveqan12d 5801	. . . 4 |- ( ( a = F /\ b = G ) -> ( ( a ` x ) R ( b ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
16	12, 15	mpteq12dv 4018	. . 3 |- ( ( a = F /\ b = G ) -> ( x e. ( dom a i^i dom b ) |-> ( ( a ` x ) R ( b ` x ) ) ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
17		df-of 5990	. . 3 |- oF R = ( a e. _V , b e. _V |-> ( x e. ( dom a i^i dom b ) |-> ( ( a ` x ) R ( b ` x ) ) ) )
18	16, 17	ovmpoga 5908	. 2 |- ( ( F e. _V /\ G e. _V /\ ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
19	2, 4, 9, 18	syl3anc 1217	1 |- ( ( F e. V /\ G e. W ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   i^i cin 3075   |-> cmpt 3997   dom cdm 4547   ` cfv 5131  (class class class)co 5782   oFcof 5988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990

This theorem is referenced by:  offres  6041