Theorem ofval 7664

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 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 7662	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9	8	fveq1d 6860	. . 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 6858	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12		fveq2 6858	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
13	11, 12	oveq12d 7405	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14		eqid 2729	. . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15		ovex 7420	. . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V
16	13, 14, 15	fvmpt 6968	. . 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 4200	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
19	5, 18	eqsstrri 3994	. . . . 5 ⊢ 𝑆 ⊆ 𝐴
20	19	sseli 3942	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴)
21		ofval.6	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	20, 21	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4201	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 3994	. . . . 5 ⊢ 𝑆 ⊆ 𝐵
25	24	sseli 3942	. . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)
26		ofval.7	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	25, 26	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	22, 27	oveq12d 7405	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷))
29	10, 17, 28	3eqtrd 2768	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   ↦ cmpt 5188   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by:  fnfvof  7670  offveq  7679  ofc1  7681  ofc2  7682  suppofss1d  8183  suppofss2d  8184  ofsubeq0  12183  ofnegsub  12184  ofsubge0  12185  seqof  14024  o1of2  15579  mndpsuppss  18692  gsumzaddlem  19851  pwspjmhmmgpd  20237  psrbagcon  21834  psrbagleadd1  21837  psrbagconf1o  21838  psrdi  21874  psrdir  21875  mplsubglem  21908  psdmplcl  22049  psdadd  22050  psdmul  22053  psdmvr  22056  matplusgcell  22320  matsubgcell  22321  rrxcph  25292  mbfaddlem  25561  i1faddlem  25594  i1fmullem  25595  itg1lea  25613  mbfi1flimlem  25623  itg2split  25650  itg2monolem1  25651  itg2addlem  25659  dvaddbr  25840  dvmulbr  25841  dvmulbrOLD  25842  plyaddlem1  26118  coeeulem  26129  coeaddlem  26154  dgradd2  26174  dgrcolem2  26180  ofmulrt  26189  plydivlem3  26203  plydivlem4  26204  plydiveu  26206  plyrem  26213  vieta1lem2  26219  elqaalem3  26229  qaa  26231  basellem7  26997  basellem9  26999  elrgspnlem1  33193  ply1degltdimlem  33618  circlemethhgt  34634  poimirlem1  37615  poimirlem2  37616  poimirlem6  37620  poimirlem7  37621  poimirlem10  37624  poimirlem11  37625  poimirlem12  37626  poimirlem17  37631  poimirlem20  37634  poimirlem23  37637  poimirlem29  37643  poimirlem31  37645  poimirlem32  37646  broucube  37648  itg2addnclem3  37667  itg2addnc  37668  ftc1anclem5  37691  lfladdcl  39064  ldualvaddval  39124  ofun  42224  mplmapghm  42544  fsuppind  42578  dgrsub2  43124  mpaaeu  43139  caofcan  44312  ofmul12  44314  ofdivrec  44315  ofdivcan4  44316  ofdivdiv2  44317  binomcxplemrat  44339  binomcxplemnotnn0  44345  amgmwlem  49791