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Theorem offval3 7965
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 3487 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 480 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
3 elex 3487 . . 3 (𝐺𝑊𝐺 ∈ V)
43adantl 481 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
5 dmexg 7890 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 inex1g 5312 . . . 4 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
7 mptexg 7217 . . . 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 5896 . . . . 5 (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹)
11 dmeq 5896 . . . . 5 (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺)
1210, 11ineqan12d 4209 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺))
13 fveq1 6883 . . . . 5 (𝑎 = 𝐹 → (𝑎𝑥) = (𝐹𝑥))
14 fveq1 6883 . . . . 5 (𝑏 = 𝐺 → (𝑏𝑥) = (𝐺𝑥))
1513, 14oveqan12d 7423 . . . 4 ((𝑎 = 𝐹𝑏 = 𝐺) → ((𝑎𝑥)𝑅(𝑏𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
1612, 15mpteq12dv 5232 . . 3 ((𝑎 = 𝐹𝑏 = 𝐺) → (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
17 df-of 7666 . . 3 f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎𝑥)𝑅(𝑏𝑥))))
1816, 17ovmpoga 7557 . 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 1368 1 ((𝐹𝑉𝐺𝑊) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cin 3942  cmpt 5224  dom cdm 5669  cfv 6536  (class class class)co 7404  f cof 7664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666
This theorem is referenced by:  offres  7966  offsplitfpar  8102  ofco2  22303  dvsinax  45183  dvcosax  45196  fdivval  47482
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