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Theorem offval 7706
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (𝜑𝐹 Fn 𝐴)
offval.2 (𝜑𝐺 Fn 𝐵)
offval.3 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
offval.6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
offval.7 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
Assertion
Ref Expression
offval (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem offval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
2 offval.3 . . . 4 (𝜑𝐴𝑉)
3 fnex 7237 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 584 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 7237 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 584 . . 3 (𝜑𝐺 ∈ V)
91fndmd 6673 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
105fndmd 6673 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
119, 10ineq12d 4221 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
12 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
1311, 12eqtrdi 2793 . . . . 5 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1413mpteq1d 5237 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
15 inex1g 5319 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
1612, 15eqeltrrid 2846 . . . . 5 (𝐴𝑉𝑆 ∈ V)
17 mptexg 7241 . . . . 5 (𝑆 ∈ V → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
182, 16, 173syl 18 . . . 4 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
1914, 18eqeltrd 2841 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
20 dmeq 5914 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21 dmeq 5914 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2220, 21ineqan12d 4222 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
23 fveq1 6905 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
24 fveq1 6905 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
2523, 24oveqan12d 7450 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
2622, 25mpteq12dv 5233 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
27 df-of 7697 . . . 4 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
2826, 27ovmpoga 7587 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
294, 8, 19, 28syl3anc 1373 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
3012eleq2i 2833 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝑆)
31 elin 3967 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3230, 31bitr3i 277 . . . 4 (𝑥𝑆 ↔ (𝑥𝐴𝑥𝐵))
33 offval.6 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3433adantrr 717 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = 𝐶)
35 offval.7 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3635adantrl 716 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐺𝑥) = 𝐷)
3734, 36oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3832, 37sylan2b 594 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3938mpteq2dva 5242 . 2 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
4029, 14, 393eqtrd 2781 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  cmpt 5225  dom cdm 5685   Fn wfn 6556  cfv 6561  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  ofval  7708  offn  7710  offval2f  7712  off  7715  ofres  7716  offval2  7717  coof  7721  ofco  7722  offveqb  7724  suppssof1  8224  o1rlimmul  15655  frlmipval  21799  frlmphllem  21800  frlmphl  21801  gsumbagdiaglem  21950  psrascl  21999  evlslem1  22106  mhpmulcl  22153  psdmplcl  22166  psdadd  22167  psdmul  22170  psrplusgpropd  22237  evls1fpws  22373  mat1dimscm  22481  rrxcph  25426  rrxds  25427  mbfadd  25696  mbfsub  25697  mbfmullem2  25759  mbfmul  25761  bddmulibl  25874  dvcmulf  25982  ofrn2  32650  off2  32651  ofresid  32652  islinds5  33395  ellspds  33396  ply1gsumz  33619  ofcof  34108  plymul02  34561  signsplypnf  34565  signsply0  34566  matunitlindflem1  37623  matunitlindflem2  37624  poimirlem3  37630  poimirlem4  37631  poimirlem16  37643  poimirlem19  37646  poimirlem28  37655  broucube  37661  itg2addnc  37681  ftc1anclem8  37707  evlsvvval  42573  dflinc2  48327  fdivmpt  48461
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