MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offval3 Structured version   Visualization version   GIF version

Theorem offval3 7825
Description: General value of (𝐹f 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
offval3 ((𝐹𝑉𝐺𝑊) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑉   𝑥,𝑊   𝑥,𝑅

Proof of Theorem offval3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3450 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 481 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3450 . . 3 (𝐺𝑊𝐺 ∈ V)
43adantl 482 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 dmexg 7750 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 inex1g 5243 . . . 4 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
7 mptexg 7097 . . . 4 ((dom 𝐹 ∩ dom 𝐺) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
85, 6, 73syl 18 . . 3 (𝐹𝑉 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
98adantr 481 . 2 ((𝐹𝑉𝐺𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
10 dmeq 5812 . . . . 5 (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹)
11 dmeq 5812 . . . . 5 (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺)
1210, 11ineqan12d 4148 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺))
13 fveq1 6773 . . . . 5 (𝑎 = 𝐹 → (𝑎𝑥) = (𝐹𝑥))
14 fveq1 6773 . . . . 5 (𝑏 = 𝐺 → (𝑏𝑥) = (𝐺𝑥))
1513, 14oveqan12d 7294 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → ((𝑎𝑥)𝑅(𝑏𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
1612, 15mpteq12dv 5165 . . 3 ((𝑎 = 𝐹𝑏 = 𝐺) → (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
17 df-of 7533 . . 3 f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))))
1816, 17ovmpoga 7427 . 2 ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
192, 4, 9, 18syl3anc 1370 1 ((𝐹𝑉𝐺𝑊) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  cmpt 5157  dom cdm 5589  cfv 6433  (class class class)co 7275  f cof 7531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533
This theorem is referenced by:  offres  7826  offsplitfpar  7960  ofco2  21600  dvsinax  43454  dvcosax  43467  fdivval  45885
  Copyright terms: Public domain W3C validator