Theorem offval3 6340

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 2827	. . 3 |- ( F e. V -> F e. _V )
2	1	adantr 276	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
3		elex 2827	. . 3 |- ( G e. W -> G e. _V )
4	3	adantl 277	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
5		dmexg 5026	. . . 4 |- ( F e. V -> dom F e. _V )
6		inex1g 4251	. . . 4 |- ( dom F e. _V -> ( dom F i^i dom G ) e. _V )
7		mptexg 5916	. . . 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 276	. 2 |- ( ( F e. V /\ G e. W ) -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
10		dmeq 4961	. . . . 5 |- ( a = F -> dom a = dom F )
11		dmeq 4961	. . . . 5 |- ( b = G -> dom b = dom G )
12	10, 11	ineqan12d 3428	. . . 4 |- ( ( a = F /\ b = G ) -> ( dom a i^i dom b ) = ( dom F i^i dom G ) )
13		fveq1 5674	. . . . 5 |- ( a = F -> ( a ` x ) = ( F ` x ) )
14		fveq1 5674	. . . . 5 |- ( b = G -> ( b ` x ) = ( G ` x ) )
15	13, 14	oveqan12d 6077	. . . 4 |- ( ( a = F /\ b = G ) -> ( ( a ` x ) R ( b ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
16	12, 15	mpteq12dv 4197	. . 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 6275	. . 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 6191	. 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 1274	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 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   i^i cin 3213   |-> cmpt 4176   dom cdm 4754   ` cfv 5357  (class class class)co 6058   oFcof 6273

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275

This theorem is referenced by:  offres  6341