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Theorem offval 7408
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 6972 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 586 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6972 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 586 . . 3 (𝜑𝐺 ∈ V)
91fndmd 6449 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
105fndmd 6449 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
119, 10ineq12d 4188 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
12 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
1311, 12syl6eq 2870 . . . . 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 2916 . . . . 5 (𝐴𝑉𝑆 ∈ V)
17 mptexg 6976 . . . . 5 (𝑆 ∈ V → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
182, 16, 173syl 18 . . . 4 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) ∈ V)
1914, 18eqeltrd 2911 . . 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 4189 . . . . 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 7167 . . . . 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 7401 . . . 4 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
2826, 27ovmpoga 7296 . . 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 1366 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
3012eleq2i 2902 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝑆)
31 elin 4167 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3230, 31bitr3i 279 . . . 4 (𝑥𝑆 ↔ (𝑥𝐴𝑥𝐵))
33 offval.6 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3433adantrr 715 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐹𝑥) = 𝐶)
35 offval.7 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3635adantrl 714 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → (𝐺𝑥) = 𝐷)
3734, 36oveq12d 7166 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑥𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3832, 37sylan2b 595 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝐶𝑅𝐷))
3938mpteq2dva 5152 . 2 (𝜑 → (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
4029, 14, 393eqtrd 2858 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  cin 3933  cmpt 5137  dom cdm 5548   Fn wfn 6343  cfv 6348  (class class class)co 7148  f cof 7399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-oprab 7152  df-mpo 7153  df-of 7401
This theorem is referenced by:  ofval  7410  offn  7412  offval2f  7413  off  7416  ofres  7417  offval2  7418  ofco  7421  offveqb  7423  suppssof1  7855  o1rlimmul  14967  gsumbagdiaglem  20147  evlslem1  20287  psrplusgpropd  20396  frlmipval  20915  frlmphllem  20916  frlmphl  20917  mat1dimscm  21076  rrxcph  23987  rrxds  23988  mbfadd  24254  mbfsub  24255  mbfmullem2  24317  mbfmul  24319  bddmulibl  24431  dvcmulf  24534  ofrn2  30379  off2  30380  ofresid  30381  islinds5  30925  ellspds  30926  ofcof  31359  plymul02  31809  signsplypnf  31813  signsply0  31814  matunitlindflem1  34880  matunitlindflem2  34881  poimirlem3  34887  poimirlem4  34888  poimirlem16  34900  poimirlem19  34903  poimirlem28  34912  broucube  34918  itg2addnc  34938  ftc1anclem8  34966  dflinc2  44455  fdivmpt  44590
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