Theorem ofval 7725

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 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 7723	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 6922	. . 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 6920	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
13	11, 12	oveq12d 7466	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15		ovex 7481	. . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V
16	13, 14, 15	fvmpt 7029	. . 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 4258	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
19	5, 18	eqsstrri 4044	. . . . 5 ⊢ 𝑆 ⊆ 𝐴
20	19	sseli 4004	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
21		ofval.6	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	20, 21	sylan2 592	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4259	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 4044	. . . . 5 ⊢ 𝑆 ⊆ 𝐵
25	24	sseli 4004	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
26		ofval.7	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	25, 26	sylan2 592	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	22, 27	oveq12d 7466	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
29	10, 17, 28	3eqtrd 2784	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ↦ cmpt 5249   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714

This theorem is referenced by:  fnfvof  7731  offveq  7739  ofc1  7741  ofc2  7742  suppofss1d  8245  suppofss2d  8246  ofsubeq0  12290  ofnegsub  12291  ofsubge0  12292  seqof  14110  o1of2  15659  gsumzaddlem  19963  pwspjmhmmgpd  20351  psrbagcon  21968  psrbagleadd1  21971  psrbagconf1o  21972  psrdi  22008  psrdir  22009  mplsubglem  22042  psdmplcl  22189  psdadd  22190  psdmul  22193  matplusgcell  22460  matsubgcell  22461  rrxcph  25445  mbfaddlem  25714  i1faddlem  25747  i1fmullem  25748  itg1lea  25767  mbfi1flimlem  25777  itg2split  25804  itg2monolem1  25805  itg2addlem  25813  dvaddbr  25994  dvmulbr  25995  dvmulbrOLD  25996  plyaddlem1  26272  coeeulem  26283  coeaddlem  26308  dgradd2  26328  dgrcolem2  26334  ofmulrt  26341  plydivlem3  26355  plydivlem4  26356  plydiveu  26358  plyrem  26365  vieta1lem2  26371  elqaalem3  26381  qaa  26383  basellem7  27148  basellem9  27150  ply1degltdimlem  33635  circlemethhgt  34620  poimirlem1  37581  poimirlem2  37582  poimirlem6  37586  poimirlem7  37587  poimirlem10  37590  poimirlem11  37591  poimirlem12  37592  poimirlem17  37597  poimirlem20  37600  poimirlem23  37603  poimirlem29  37609  poimirlem31  37611  poimirlem32  37612  broucube  37614  itg2addnclem3  37633  itg2addnc  37634  ftc1anclem5  37657  lfladdcl  39027  ldualvaddval  39087  ofun  42231  mplmapghm  42511  fsuppind  42545  dgrsub2  43092  mpaaeu  43107  caofcan  44292  ofmul12  44294  ofdivrec  44295  ofdivcan4  44296  ofdivdiv2  44297  binomcxplemrat  44319  binomcxplemnotnn0  44325  mndpsuppss  48096  amgmwlem  48896