Theorem offval3 6305

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 2815	. . 3 |- ( F e. V -> F e. _V )
2	1	adantr 276	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
3		elex 2815	. . 3 |- ( G e. W -> G e. _V )
4	3	adantl 277	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
5		dmexg 5002	. . . 4 |- ( F e. V -> dom F e. _V )
6		inex1g 4230	. . . 4 |- ( dom F e. _V -> ( dom F i^i dom G ) e. _V )
7		mptexg 5889	. . . 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 4937	. . . . 5 |- ( a = F -> dom a = dom F )
11		dmeq 4937	. . . . 5 |- ( b = G -> dom b = dom G )
12	10, 11	ineqan12d 3412	. . . 4 |- ( ( a = F /\ b = G ) -> ( dom a i^i dom b ) = ( dom F i^i dom G ) )
13		fveq1 5647	. . . . 5 |- ( a = F -> ( a ` x ) = ( F ` x ) )
14		fveq1 5647	. . . . 5 |- ( b = G -> ( b ` x ) = ( G ` x ) )
15	13, 14	oveqan12d 6047	. . . 4 |- ( ( a = F /\ b = G ) -> ( ( a ` x ) R ( b ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
16	12, 15	mpteq12dv 4176	. . 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 6244	. . 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 6161	. 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 2202   _Vcvv 2803   i^i cin 3200   |-> cmpt 4155   dom cdm 4731   ` cfv 5333  (class class class)co 6028   oFcof 6242

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244

This theorem is referenced by:  offres  6306