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Theorem ofval 6948
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 2652 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2652 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 6946 . . . 4 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 6231 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 480 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 fveq2 6229 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
12 fveq2 6229 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1311, 12oveq12d 6708 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
14 eqid 2651 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
15 ovex 6718 . . . 4 ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V
1613, 14, 15fvmpt 6321 . . 3 (𝑋𝑆 → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
1716adantl 481 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
18 inss1 3866 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
195, 18eqsstr3i 3669 . . . . 5 𝑆𝐴
2019sseli 3632 . . . 4 (𝑋𝑆𝑋𝐴)
21 ofval.6 . . . 4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2220, 21sylan2 490 . . 3 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3867 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstr3i 3669 . . . . 5 𝑆𝐵
2524sseli 3632 . . . 4 (𝑋𝑆𝑋𝐵)
26 ofval.7 . . . 4 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2725, 26sylan2 490 . . 3 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2822, 27oveq12d 6708 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
2910, 17, 283eqtrd 2689 1 ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cin 3606  cmpt 4762   Fn wfn 5921  cfv 5926  (class class class)co 6690  𝑓 cof 6937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939
This theorem is referenced by:  fnfvof  6953  offveq  6960  ofc1  6962  ofc2  6963  suppofss1d  7377  suppofss2d  7378  ofsubeq0  11055  ofnegsub  11056  ofsubge0  11057  seqof  12898  o1of2  14387  gsumzaddlem  18367  psrbagcon  19419  psrbagconf1o  19422  psrdi  19454  psrdir  19455  mplsubglem  19482  matplusgcell  20287  matsubgcell  20288  rrxcph  23226  mbfaddlem  23472  i1faddlem  23505  i1fmullem  23506  itg1lea  23524  mbfi1flimlem  23534  itg2split  23561  itg2monolem1  23562  itg2addlem  23570  dvaddbr  23746  dvmulbr  23747  plyaddlem1  24014  coeeulem  24025  coeaddlem  24050  dgradd2  24069  dgrcolem2  24075  ofmulrt  24082  plydivlem3  24095  plydivlem4  24096  plydiveu  24098  plyrem  24105  vieta1lem2  24111  elqaalem3  24121  qaa  24123  basellem7  24858  basellem9  24860  circlemethhgt  30849  poimirlem1  33540  poimirlem2  33541  poimirlem6  33545  poimirlem7  33546  poimirlem10  33549  poimirlem11  33550  poimirlem12  33551  poimirlem17  33556  poimirlem20  33559  poimirlem23  33562  poimirlem29  33568  poimirlem31  33570  poimirlem32  33571  broucube  33573  itg2addnclem3  33593  itg2addnc  33594  ftc1anclem5  33619  lfladdcl  34676  ldualvaddval  34736  dgrsub2  38022  mpaaeu  38037  caofcan  38839  ofmul12  38841  ofdivrec  38842  ofdivcan4  38843  ofdivdiv2  38844  binomcxplemrat  38866  binomcxplemnotnn0  38872  mndpsuppss  42477  amgmwlem  42876
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