Theorem ofvalg 6070

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)

Hypotheses

Ref	Expression
offval.1	|- ( ph -> F Fn A )
offval.2	|- ( ph -> G Fn B )
offval.3	|- ( ph -> A e. V )
offval.4	|- ( ph -> B e. W )
offval.5	|- ( A i^i B ) = S
ofval.6	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofval.7	|- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
ofval.8	|- ( ( ph /\ X e. S ) -> ( C R D ) e. U )

Assertion

Ref	Expression
ofvalg	|- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Proof of Theorem ofvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . 5 |- ( ph -> F Fn A )
2		offval.2	. . . . 5 |- ( ph -> G Fn B )
3		offval.3	. . . . 5 |- ( ph -> A e. V )
4		offval.4	. . . . 5 |- ( ph -> B e. W )
5		offval.5	. . . . 5 |- ( A i^i B ) = S
6		eqidd 2171	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2171	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6068	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5498	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 274	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		eqid 2170	. . 3 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
12		fveq2 5496	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
13		fveq2 5496	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
14	12, 13	oveq12d 5871	. . 3 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
15		simpr 109	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
16		inss1 3347	. . . . . . . 8 |- ( A i^i B ) C_ A
17	5, 16	eqsstrri 3180	. . . . . . 7 |- S C_ A
18	17	sseli 3143	. . . . . 6 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . 6 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 284	. . . . 5 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21		inss2 3348	. . . . . . . 8 |- ( A i^i B ) C_ B
22	5, 21	eqsstrri 3180	. . . . . . 7 |- S C_ B
23	22	sseli 3143	. . . . . 6 |- ( X e. S -> X e. B )
24		ofval.7	. . . . . 6 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
25	23, 24	sylan2 284	. . . . 5 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
26	20, 25	oveq12d 5871	. . . 4 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) = ( C R D ) )
27		ofval.8	. . . 4 |- ( ( ph /\ X e. S ) -> ( C R D ) e. U )
28	26, 27	eqeltrd 2247	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. U )
29	11, 14, 15, 28	fvmptd3 5589	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
30	10, 29, 26	3eqtrd 2207	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   i^i cin 3120   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   oFcof 6059

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061

This theorem is referenced by:  offeq  6074  dvaddxxbr  13459  dvmulxxbr  13460