Theorem ofval 7642

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 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 7640	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 6842	. . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
10	9	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋))
11		fveq2 6840	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12		fveq2 6840	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
13	11, 12	oveq12d 7385	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15		ovex 7400	. . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V
16	13, 14, 15	fvmpt 6947	. . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
17	16	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
18		inss1 4177	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
19	5, 18	eqsstrri 3969	. . . . 5 ⊢ 𝑆 ⊆ 𝐴
20	19	sseli 3917	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
21		ofval.6	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	20, 21	sylan2 594	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4178	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 3969	. . . . 5 ⊢ 𝑆 ⊆ 𝐵
25	24	sseli 3917	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
26		ofval.7	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	25, 26	sylan2 594	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	22, 27	oveq12d 7385	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
29	10, 17, 28	3eqtrd 2775	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   ↦ cmpt 5166   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631

This theorem is referenced by:  fnfvof  7648  offveq  7657  ofc1  7659  ofc2  7660  suppofss1d  8154  suppofss2d  8155  ofsubeq0  12156  ofnegsub  12157  ofsubge0  12158  seqof  14021  o1of2  15575  mndpsuppss  18733  gsumzaddlem  19896  pwspjmhmmgpd  20307  psrbagcon  21905  psrbagleadd1  21908  psrbagconf1o  21909  psrdi  21943  psrdir  21944  mplsubglem  21977  psdmplcl  22128  psdadd  22129  psdmul  22132  psdmvr  22135  matplusgcell  22398  matsubgcell  22399  rrxcph  25359  mbfaddlem  25627  i1faddlem  25660  i1fmullem  25661  itg1lea  25679  mbfi1flimlem  25689  itg2split  25716  itg2monolem1  25717  itg2addlem  25725  dvaddbr  25905  dvmulbr  25906  plyaddlem1  26178  coeeulem  26189  coeaddlem  26214  dgradd2  26233  dgrcolem2  26239  ofmulrt  26248  plydivlem3  26261  plydivlem4  26262  plydiveu  26264  plyrem  26271  vieta1lem2  26277  elqaalem3  26287  qaa  26289  basellem7  27050  basellem9  27052  elrgspnlem1  33303  ply1degltdimlem  33766  circlemethhgt  34787  poimirlem1  37942  poimirlem2  37943  poimirlem6  37947  poimirlem7  37948  poimirlem10  37951  poimirlem11  37952  poimirlem12  37953  poimirlem17  37958  poimirlem20  37961  poimirlem23  37964  poimirlem29  37970  poimirlem31  37972  poimirlem32  37973  broucube  37975  itg2addnclem3  37994  itg2addnc  37995  ftc1anclem5  38018  lfladdcl  39517  ldualvaddval  39577  ofun  42677  mplmapghm  42997  fsuppind  43023  dgrsub2  43563  mpaaeu  43578  caofcan  44750  ofmul12  44752  ofdivrec  44753  ofdivcan4  44754  ofdivdiv2  44755  binomcxplemrat  44777  binomcxplemnotnn0  44783  amgmwlem  50277