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Theorem ofval 7140
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 2800 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2800 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7138 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 6413 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 473 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 fveq2 6411 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
12 fveq2 6411 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1311, 12oveq12d 6896 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
14 eqid 2799 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
15 ovex 6910 . . . 4 ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V
1613, 14, 15fvmpt 6507 . . 3 (𝑋𝑆 → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
1716adantl 474 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
18 inss1 4028 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
195, 18eqsstr3i 3832 . . . . 5 𝑆𝐴
2019sseli 3794 . . . 4 (𝑋𝑆𝑋𝐴)
21 ofval.6 . . . 4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2220, 21sylan2 587 . . 3 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 4029 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstr3i 3832 . . . . 5 𝑆𝐵
2524sseli 3794 . . . 4 (𝑋𝑆𝑋𝐵)
26 ofval.7 . . . 4 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2725, 26sylan2 587 . . 3 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2822, 27oveq12d 6896 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
2910, 17, 283eqtrd 2837 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cin 3768  cmpt 4922   Fn wfn 6096  cfv 6101  (class class class)co 6878  𝑓 cof 7129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131
This theorem is referenced by:  fnfvof  7145  offveq  7152  ofc1  7154  ofc2  7155  suppofss1d  7570  suppofss2d  7571  ofsubeq0  11309  ofnegsub  11310  ofsubge0  11311  seqof  13112  o1of2  14684  gsumzaddlem  18636  psrbagcon  19694  psrbagconf1o  19697  psrdi  19729  psrdir  19730  mplsubglem  19757  matplusgcell  20564  matsubgcell  20565  rrxcph  23518  mbfaddlem  23768  i1faddlem  23801  i1fmullem  23802  itg1lea  23820  mbfi1flimlem  23830  itg2split  23857  itg2monolem1  23858  itg2addlem  23866  dvaddbr  24042  dvmulbr  24043  plyaddlem1  24310  coeeulem  24321  coeaddlem  24346  dgradd2  24365  dgrcolem2  24371  ofmulrt  24378  plydivlem3  24391  plydivlem4  24392  plydiveu  24394  plyrem  24401  vieta1lem2  24407  elqaalem3  24417  qaa  24419  basellem7  25165  basellem9  25167  circlemethhgt  31241  poimirlem1  33899  poimirlem2  33900  poimirlem6  33904  poimirlem7  33905  poimirlem10  33908  poimirlem11  33909  poimirlem12  33910  poimirlem17  33915  poimirlem20  33918  poimirlem23  33921  poimirlem29  33927  poimirlem31  33929  poimirlem32  33930  broucube  33932  itg2addnclem3  33951  itg2addnc  33952  ftc1anclem5  33977  lfladdcl  35092  ldualvaddval  35152  dgrsub2  38490  mpaaeu  38505  caofcan  39304  ofmul12  39306  ofdivrec  39307  ofdivcan4  39308  ofdivdiv2  39309  binomcxplemrat  39331  binomcxplemnotnn0  39337  mndpsuppss  42951  amgmwlem  43350
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