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Theorem offval 6946
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 6522 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 694 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6522 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 694 . . 3 (𝜑𝐺 ∈ V)
9 fndm 6028 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
101, 9syl 17 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
11 fndm 6028 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
125, 11syl 17 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
1310, 12ineq12d 3848 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
14 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
1513, 14syl6eq 2701 . . . . 5 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1615mpteq1d 4771 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
17 inex1g 4834 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
1814, 17syl5eqelr 2735 . . . . 5 (𝐴𝑉𝑆 ∈ V)
19 mptexg 6525 . . . . 5 (𝑆 ∈ V → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
202, 18, 193syl 18 . . . 4 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
2116, 20eqeltrd 2730 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
22 dmeq 5356 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
23 dmeq 5356 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2422, 23ineqan12d 3849 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
25 fveq1 6228 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
26 fveq1 6228 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
2725, 26oveqan12d 6709 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
2824, 27mpteq12dv 4766 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
29 df-of 6939 . . . 4 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
3028, 29ovmpt2ga 6832 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
314, 8, 21, 30syl3anc 1366 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
3214eleq2i 2722 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝑆)
33 elin 3829 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3432, 33bitr3i 266 . . . 4 (𝑥𝑆 ↔ (𝑥𝐴𝑥𝐵))
35 offval.6 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3635adantrr 753 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = 𝐶)
37 offval.7 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3837adantrl 752 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐺𝑥) = 𝐷)
3936, 38oveq12d 6708 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
4034, 39sylan2b 491 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
4140mpteq2dva 4777 . 2 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
4231, 16, 413eqtrd 2689 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  cmpt 4762  dom cdm 5143   Fn wfn 5921  cfv 5926  (class class class)co 6690  𝑓 cof 6937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939
This theorem is referenced by:  ofval  6948  offn  6950  offval2f  6951  off  6954  ofres  6955  offval2  6956  ofco  6959  offveqb  6961  suppssof1  7373  o1rlimmul  14393  gsumbagdiaglem  19423  evlslem1  19563  psrplusgpropd  19654  frlmipval  20166  frlmphllem  20167  frlmphl  20168  mat1dimscm  20329  rrxcph  23226  rrxds  23227  mbfadd  23473  mbfsub  23474  mbfmullem2  23536  mbfmul  23538  bddmulibl  23650  dvcmulf  23753  ofrn2  29570  off2  29571  ofresid  29572  ofcof  30297  plymul02  30751  signsplypnf  30755  signsply0  30756  matunitlindflem1  33535  matunitlindflem2  33536  poimirlem3  33542  poimirlem4  33543  poimirlem16  33555  poimirlem19  33558  poimirlem28  33567  broucube  33573  itg2addnc  33594  ftc1anclem8  33622  dflinc2  42524  fdivmpt  42659
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