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Theorem offval0 42778
 Description: Value of an operation applied to two functions. (Contributed by AV, 15-May-2020.)
Assertion
Ref Expression
offval0 ((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑅
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem offval0
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 7050 . . 3 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
21a1i 11 . 2 ((𝐹𝑉𝐺𝑊) → ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))))
3 dmeq 5467 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5467 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
53, 4ineqan12d 3947 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6 fveq1 6339 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 6339 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
86, 7oveqan12d 6820 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
95, 8mpteq12dv 4873 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
109adantl 473 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
11 elex 3340 . . 3 (𝐹𝑉𝐹 ∈ V)
1211adantr 472 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
13 elex 3340 . . 3 (𝐺𝑊𝐺 ∈ V)
1413adantl 473 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
15 dmexg 7250 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
1615adantr 472 . . 3 ((𝐹𝑉𝐺𝑊) → dom 𝐹 ∈ V)
17 inex1g 4941 . . 3 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
18 mptexg 6636 . . 3 ((dom 𝐹 ∩ dom 𝐺) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
1916, 17, 183syl 18 . 2 ((𝐹𝑉𝐺𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
202, 10, 12, 14, 19ovmpt2d 6941 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 1620   ∈ wcel 2127  Vcvv 3328   ∩ cin 3702   ↦ cmpt 4869  dom cdm 5254  ‘cfv 6037  (class class class)co 6801   ↦ cmpt2 6803   ∘𝑓 cof 7048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-of 7050 This theorem is referenced by:  fdivval  42812
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