Theorem ofval 7687

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
offval.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
offval.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
ofval.6	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
ofval.7	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)

Assertion

Ref	Expression
ofval	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem ofval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offval.2	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		offval.3	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		offval.4	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		offval.5	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = 𝑆
6		eqidd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 7685	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 6885	. . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
10	9	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
11		fveq2 6883	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12		fveq2 6883	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
13	11, 12	oveq12d 7430	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15		ovex 7445	. . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V
16	13, 14, 15	fvmpt 6991	. . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
17	16	adantl 486	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
18		inss1 4190	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
19	5, 18	eqsstrri 3985	. . . . 5 ⊢ 𝑆 ⊆ 𝐴
20	19	sseli 3934	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
21		ofval.6	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	20, 21	sylan2 604	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4191	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 3985	. . . . 5 ⊢ 𝑆 ⊆ 𝐵
25	24	sseli 3934	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
26		ofval.7	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	25, 26	sylan2 604	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	22, 27	oveq12d 7430	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
29	10, 17, 28	3eqtrd 2802	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∩ cin 3905   ↦ cmpt 5193   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   ∘_f cof 7674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676

This theorem is referenced by:  fnfvof  7693  offveq  7702  ofc1  7704  ofc2  7705  suppofss1d  8201  suppofss2d  8202  ofsubeq0  12216  ofnegsub  12217  ofsubge0  12218  seqof  14097  o1of2  15666  mndpsuppss  18824  gsumzaddlem  19992  pwspjmhmmgpd  20410  psrbagcon  22056  psrbagleadd1  22059  psrbagconf1o  22060  psrdi  22095  psrdir  22096  mplsubglem  22129  mplmapghm  22254  psdmplcl  22306  psdadd  22307  psdmul  22310  psdmvr  22313  matplusgcell  22571  matsubgcell  22572  rrxcph  25532  mbfaddlem  25800  i1faddlem  25833  i1fmullem  25834  itg1lea  25852  mbfi1flimlem  25862  itg2split  25889  itg2monolem1  25890  itg2addlem  25898  dvaddbr  26078  dvmulbr  26079  plyaddlem1  26351  coeeulem  26362  coeaddlem  26387  dgradd2  26406  dgrcolem2  26412  ofmulrt  26421  plydivlem3  26437  plydivlem4  26438  plydiveu  26440  plyrem  26447  vieta1lem2  26453  elqaalem3  26463  qaa  26465  basellem7  27232  basellem9  27234  elrgspnlem1  33543  0mplrim  33885  selvply1rhmlemb  33890  selvply1rhmlem4  33894  ply1degltdimlem  33993  circlemethhgt  35011  poimirlem1  38253  poimirlem2  38254  poimirlem6  38258  poimirlem7  38259  poimirlem10  38262  poimirlem11  38263  poimirlem12  38264  poimirlem17  38269  poimirlem20  38272  poimirlem23  38275  poimirlem29  38281  poimirlem31  38283  poimirlem32  38284  broucube  38286  itg2addnclem3  38305  itg2addnc  38306  ftc1anclem5  38329  lfladdcl  39826  ldualvaddval  39886  ofun  42987  fsuppind  43305  dgrsub2  43845  mpaaeu  43860  caofcan  45016  ofmul12  45018  ofdivrec  45019  ofdivcan4  45020  ofdivdiv2  45021  binomcxplemrat  45043  binomcxplemnotnn0  45049  amgmwlem  50585