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Theorem ofval 7071
Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (𝜑𝐹 Fn 𝐴)
offval.2 (𝜑𝐺 Fn 𝐵)
offval.3 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
ofval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofval.7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
Assertion
Ref Expression
ofval ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem ofval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . 5 (𝜑𝐹 Fn 𝐴)
2 offval.2 . . . . 5 (𝜑𝐺 Fn 𝐵)
3 offval.3 . . . . 5 (𝜑𝐴𝑉)
4 offval.4 . . . . 5 (𝜑𝐵𝑊)
5 offval.5 . . . . 5 (𝐴𝐵) = 𝑆
6 eqidd 2761 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2761 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7069 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 6354 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 472 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 fveq2 6352 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
12 fveq2 6352 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1311, 12oveq12d 6831 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
14 eqid 2760 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
15 ovex 6841 . . . 4 ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V
1613, 14, 15fvmpt 6444 . . 3 (𝑋𝑆 → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
1716adantl 473 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
18 inss1 3976 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
195, 18eqsstr3i 3777 . . . . 5 𝑆𝐴
2019sseli 3740 . . . 4 (𝑋𝑆𝑋𝐴)
21 ofval.6 . . . 4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2220, 21sylan2 492 . . 3 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3977 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstr3i 3777 . . . . 5 𝑆𝐵
2524sseli 3740 . . . 4 (𝑋𝑆𝑋𝐵)
26 ofval.7 . . . 4 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2725, 26sylan2 492 . . 3 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2822, 27oveq12d 6831 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
2910, 17, 283eqtrd 2798 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cin 3714  cmpt 4881   Fn wfn 6044  cfv 6049  (class class class)co 6813  𝑓 cof 7060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062
This theorem is referenced by:  fnfvof  7076  offveq  7083  ofc1  7085  ofc2  7086  suppofss1d  7501  suppofss2d  7502  ofsubeq0  11209  ofnegsub  11210  ofsubge0  11211  seqof  13052  o1of2  14542  gsumzaddlem  18521  psrbagcon  19573  psrbagconf1o  19576  psrdi  19608  psrdir  19609  mplsubglem  19636  matplusgcell  20441  matsubgcell  20442  rrxcph  23380  mbfaddlem  23626  i1faddlem  23659  i1fmullem  23660  itg1lea  23678  mbfi1flimlem  23688  itg2split  23715  itg2monolem1  23716  itg2addlem  23724  dvaddbr  23900  dvmulbr  23901  plyaddlem1  24168  coeeulem  24179  coeaddlem  24204  dgradd2  24223  dgrcolem2  24229  ofmulrt  24236  plydivlem3  24249  plydivlem4  24250  plydiveu  24252  plyrem  24259  vieta1lem2  24265  elqaalem3  24275  qaa  24277  basellem7  25012  basellem9  25014  circlemethhgt  31030  poimirlem1  33723  poimirlem2  33724  poimirlem6  33728  poimirlem7  33729  poimirlem10  33732  poimirlem11  33733  poimirlem12  33734  poimirlem17  33739  poimirlem20  33742  poimirlem23  33745  poimirlem29  33751  poimirlem31  33753  poimirlem32  33754  broucube  33756  itg2addnclem3  33776  itg2addnc  33777  ftc1anclem5  33802  lfladdcl  34861  ldualvaddval  34921  dgrsub2  38207  mpaaeu  38222  caofcan  39024  ofmul12  39026  ofdivrec  39027  ofdivcan4  39028  ofdivdiv2  39029  binomcxplemrat  39051  binomcxplemnotnn0  39057  mndpsuppss  42662  amgmwlem  43061
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