Theorem ofvalg 6191

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 2208	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2208	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6189	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5601	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 276	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		eqid 2207	. . 3 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
12		fveq2 5599	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
13		fveq2 5599	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
14	12, 13	oveq12d 5985	. . 3 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
15		simpr 110	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
16		inss1 3401	. . . . . . . 8 |- ( A i^i B ) C_ A
17	5, 16	eqsstrri 3234	. . . . . . 7 |- S C_ A
18	17	sseli 3197	. . . . . 6 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . 6 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 286	. . . . 5 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21		inss2 3402	. . . . . . . 8 |- ( A i^i B ) C_ B
22	5, 21	eqsstrri 3234	. . . . . . 7 |- S C_ B
23	22	sseli 3197	. . . . . 6 |- ( X e. S -> X e. B )
24		ofval.7	. . . . . 6 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
25	23, 24	sylan2 286	. . . . 5 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
26	20, 25	oveq12d 5985	. . . 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 2284	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. U )
29	11, 14, 15, 28	fvmptd3 5696	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
30	10, 29, 26	3eqtrd 2244	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   i^i cin 3173   |-> cmpt 4121   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   oFcof 6179

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by:  offeq  6195  ofc1g  6203  ofc2g  6204  ofnegsub  9070  gsumfzmptfidmadd  13790  mplsubgfilemcl  14576  dvaddxxbr  15288  dvmulxxbr  15289  plyaddlem1  15334