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Theorem offval 7405
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 6971 . . . 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 6971 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 584 . . 3 (𝜑𝐺 ∈ V)
91fndmd 6449 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
105fndmd 6449 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
119, 10ineq12d 4187 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
12 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
1311, 12syl6eq 2869 . . . . 5 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
1413mpteq1d 5146 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
15 inex1g 5214 . . . . . 6 (𝐴𝑉 → (𝐴𝐵) ∈ V)
1612, 15eqeltrrid 2915 . . . . 5 (𝐴𝑉𝑆 ∈ V)
17 mptexg 6975 . . . . 5 (𝑆 ∈ V → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
182, 16, 173syl 18 . . . 4 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
1914, 18eqeltrd 2910 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
20 dmeq 5765 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21 dmeq 5765 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2220, 21ineqan12d 4188 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
23 fveq1 6662 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
24 fveq1 6662 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
2523, 24oveqan12d 7164 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝐹𝑥)𝑅(𝐺𝑥)))
2622, 25mpteq12dv 5142 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
27 df-of 7398 . . . 4 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
2826, 27ovmpoga 7293 . . 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 1363 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
3012eleq2i 2901 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝑆)
31 elin 4166 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3230, 31bitr3i 278 . . . 4 (𝑥𝑆 ↔ (𝑥𝐴𝑥𝐵))
33 offval.6 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3433adantrr 713 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = 𝐶)
35 offval.7 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3635adantrl 712 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐺𝑥) = 𝐷)
3734, 36oveq12d 7163 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3832, 37sylan2b 593 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3938mpteq2dva 5152 . 2 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
4029, 14, 393eqtrd 2857 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  cmpt 5137  dom cdm 5548   Fn wfn 6343  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  ofval  7407  offn  7409  offval2f  7410  off  7413  ofres  7414  offval2  7415  ofco  7418  offveqb  7420  suppssof1  7852  o1rlimmul  14963  gsumbagdiaglem  20083  evlslem1  20223  psrplusgpropd  20332  frlmipval  20851  frlmphllem  20852  frlmphl  20853  mat1dimscm  21012  rrxcph  23922  rrxds  23923  mbfadd  24189  mbfsub  24190  mbfmullem2  24252  mbfmul  24254  bddmulibl  24366  dvcmulf  24469  ofrn2  30315  off2  30316  ofresid  30317  islinds5  30859  ellspds  30860  ofcof  31265  plymul02  31715  signsplypnf  31719  signsply0  31720  matunitlindflem1  34769  matunitlindflem2  34770  poimirlem3  34776  poimirlem4  34777  poimirlem16  34789  poimirlem19  34792  poimirlem28  34801  broucube  34807  itg2addnc  34827  ftc1anclem8  34855  dflinc2  44393  fdivmpt  44528
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