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Theorem offval 6802
 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 6386 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 691 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6386 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 691 . . 3 (𝜑𝐺 ∈ V)
9 fndm 5904 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
101, 9syl 17 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
11 fndm 5904 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
125, 11syl 17 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
1310, 12ineq12d 3777 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
14 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
1513, 14syl6eq 2660 . . . . 5 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1615mpteq1d 4666 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
17 inex1g 4729 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
1814, 17syl5eqelr 2693 . . . . 5 (𝐴𝑉𝑆 ∈ V)
19 mptexg 6389 . . . . 5 (𝑆 ∈ V → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
202, 18, 193syl 18 . . . 4 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
2116, 20eqeltrd 2688 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
22 dmeq 5246 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
23 dmeq 5246 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2422, 23ineqan12d 3778 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
25 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
26 fveq1 6102 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
2725, 26oveqan12d 6568 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
2824, 27mpteq12dv 4663 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
29 df-of 6795 . . . 4 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
3028, 29ovmpt2ga 6688 . . 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 1318 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
3214eleq2i 2680 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝑆)
33 elin 3758 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3432, 33bitr3i 265 . . . 4 (𝑥𝑆 ↔ (𝑥𝐴𝑥𝐵))
35 offval.6 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3635adantrr 749 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = 𝐶)
37 offval.7 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3837adantrl 748 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐺𝑥) = 𝐷)
3936, 38oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
4034, 39sylan2b 491 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
4140mpteq2dva 4672 . 2 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
4231, 16, 413eqtrd 2648 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by:  ofval  6804  offn  6806  offval2f  6807  off  6810  ofres  6811  offval2  6812  ofco  6815  offveqb  6817  suppssof1  7215  o1rlimmul  14197  gsumbagdiaglem  19196  evlslem1  19336  psrplusgpropd  19427  frlmipval  19937  frlmphllem  19938  frlmphl  19939  mat1dimscm  20100  rrxcph  22988  rrxds  22989  mbfadd  23234  mbfsub  23235  mbfmullem2  23297  mbfmul  23299  bddmulibl  23411  dvcmulf  23514  ofrn2  28822  off2  28823  ofresid  28824  ofcof  29496  plymul02  29949  signsplypnf  29953  signsply0  29954  matunitlindflem1  32575  matunitlindflem2  32576  poimirlem3  32582  poimirlem4  32583  poimirlem16  32595  poimirlem19  32598  poimirlem28  32607  broucube  32613  itg2addnc  32634  ftc1anclem8  32662  dflinc2  41993  fdivmpt  42132
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