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Theorem offval3 7053
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 3185 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 480 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3185 . . 3 (𝐺𝑊𝐺 ∈ V)
43adantl 481 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 dmexg 6989 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 inex1g 4729 . . . 4 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
7 mptexg 6389 . . . 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 480 . 2 ((𝐹𝑉𝐺𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
10 dmeq 5246 . . . . 5 (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹)
11 dmeq 5246 . . . . 5 (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺)
1210, 11ineqan12d 3778 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺))
13 fveq1 6102 . . . . 5 (𝑎 = 𝐹 → (𝑎𝑥) = (𝐹𝑥))
14 fveq1 6102 . . . . 5 (𝑏 = 𝐺 → (𝑏𝑥) = (𝐺𝑥))
1513, 14oveqan12d 6568 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → ((𝑎𝑥)𝑅(𝑏𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
1612, 15mpteq12dv 4663 . . 3 ((𝑎 = 𝐹𝑏 = 𝐺) → (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
17 df-of 6795 . . 3 𝑓 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))))
1816, 17ovmpt2ga 6688 . 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 1318 1 ((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  cmpt 4643  dom cdm 5038  cfv 5804  (class class class)co 6549  𝑓 cof 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795
This theorem is referenced by:  offres  7054  ofco2  20076  dvsinax  38801  dvcosax  38816
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