Theorem offval3 6117

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 2742	. . 3 |- ( F e. V -> F e. _V )
2	1	adantr 274	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
3		elex 2742	. . 3 |- ( G e. W -> G e. _V )
4	3	adantl 275	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
5		dmexg 4876	. . . 4 |- ( F e. V -> dom F e. _V )
6		inex1g 4126	. . . 4 |- ( dom F e. _V -> ( dom F i^i dom G ) e. _V )
7		mptexg 5725	. . . 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 4812	. . . . 5 |- ( a = F -> dom a = dom F )
11		dmeq 4812	. . . . 5 |- ( b = G -> dom b = dom G )
12	10, 11	ineqan12d 3331	. . . 4 |- ( ( a = F /\ b = G ) -> ( dom a i^i dom b ) = ( dom F i^i dom G ) )
13		fveq1 5498	. . . . 5 |- ( a = F -> ( a ` x ) = ( F ` x ) )
14		fveq1 5498	. . . . 5 |- ( b = G -> ( b ` x ) = ( G ` x ) )
15	13, 14	oveqan12d 5876	. . . 4 |- ( ( a = F /\ b = G ) -> ( ( a ` x ) R ( b ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
16	12, 15	mpteq12dv 4072	. . 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 6065	. . 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 5986	. 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 1234	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 1349   e. wcel 2142   _Vcvv 2731   i^i cin 3121   |-> cmpt 4051   dom cdm 4612   ` cfv 5200  (class class class)co 5857   oFcof 6063

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-ov 5860  df-oprab 5861  df-mpo 5862  df-of 6065

This theorem is referenced by:  offres  6118