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Theorem offval0 42070
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 6894 . . 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 5322 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5322 . . . . 5 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
53, 4ineqan12d 3814 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6 fveq1 6188 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 6188 . . . . 5 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
86, 7oveqan12d 6666 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
95, 8mpteq12dv 4731 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
109adantl 482 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
11 elex 3210 . . 3 (𝐹𝑉𝐹 ∈ V)
1211adantr 481 . 2 ((𝐹𝑉𝐺𝑊) → 𝐹 ∈ V)
13 elex 3210 . . 3 (𝐺𝑊𝐺 ∈ V)
1413adantl 482 . 2 ((𝐹𝑉𝐺𝑊) → 𝐺 ∈ V)
15 dmexg 7094 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
1615adantr 481 . . 3 ((𝐹𝑉𝐺𝑊) → dom 𝐹 ∈ V)
17 inex1g 4799 . . 3 (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
18 mptexg 6481 . . 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 6785 1 ((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cin 3571  cmpt 4727  dom cdm 5112  cfv 5886  (class class class)co 6647  cmpt2 6649  𝑓 cof 6892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894
This theorem is referenced by:  fdivval  42104
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