Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  offval0 Structured version   Visualization version   GIF version

Theorem offval0 42198
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 6671 . . 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 5137 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5137 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
53, 4ineqan12d 3681 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6 fveq1 5986 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 5986 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
86, 7oveqan12d 6445 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
95, 8mpteq12dv 4561 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
109adantl 480 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
11 elex 3089 . . 3 (𝐹𝑉𝐹 ∈ V)
1211adantr 479 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
13 elex 3089 . . 3 (𝐺𝑊𝐺 ∈ V)
1413adantl 480 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
15 dmexg 6865 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
1615adantr 479 . . 3 ((𝐹𝑉𝐺𝑊) → dom 𝐹 ∈ V)
17 inex1g 4628 . . 3 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
18 mptexg 6266 . . 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 6563 1 ((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  cin 3443  cmpt 4541  dom cdm 4932  cfv 5689  (class class class)co 6426  cmpt2 6428  𝑓 cof 6669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-of 6671
This theorem is referenced by:  fdivval  42236
  Copyright terms: Public domain W3C validator