Theorem offval3 5761

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 2566	. . 3 |- ( F e. V -> F e. _V )
2	1	adantr 261	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
3		elex 2566	. . 3 |- ( G e. W -> G e. _V )
4	3	adantl 262	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
5		dmexg 4596	. . . 4 |- ( F e. V -> dom F e. _V )
6		inex1g 3893	. . . 4 |- ( dom F e. _V -> ( dom F i^i dom G ) e. _V )
7		mptexg 5386	. . . 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 261	. 2 |- ( ( F e. V /\ G e. W ) -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
10		dmeq 4535	. . . . 5 |- ( a = F -> dom a = dom F )
11		dmeq 4535	. . . . 5 |- ( b = G -> dom b = dom G )
12	10, 11	ineqan12d 3140	. . . 4 |- ( ( a = F /\ b = G ) -> ( dom a i^i dom b ) = ( dom F i^i dom G ) )
13		fveq1 5177	. . . . 5 |- ( a = F -> ( a ` x ) = ( F ` x ) )
14		fveq1 5177	. . . . 5 |- ( b = G -> ( b ` x ) = ( G ` x ) )
15	13, 14	oveqan12d 5531	. . . 4 |- ( ( a = F /\ b = G ) -> ( ( a ` x ) R ( b ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
16	12, 15	mpteq12dv 3839	. . 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 5712	. . 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	ovmpt2ga 5630	. 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 1135	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 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   i^i cin 2916   |-> cmpt 3818   dom cdm 4345   ` cfv 4902  (class class class)co 5512   oFcof 5710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712

This theorem is referenced by:  offres  5762