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Theorem ofcfval 34062
Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1 (𝜑𝐹 Fn 𝐴)
ofcfval.2 (𝜑𝐴𝑉)
ofcfval.3 (𝜑𝐶𝑊)
ofcfval.6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Assertion
Ref Expression
ofcfval (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofcfval
Dummy variables 𝑓 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofc 34060 . . . 4 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
21a1i 11 . . 3 (𝜑 → ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))))
3 simprl 770 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑓 = 𝐹)
43dmeqd 5930 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
53fveq1d 6922 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑓𝑥) = (𝐹𝑥))
6 simprr 772 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑐 = 𝐶)
75, 6oveq12d 7466 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
84, 7mpteq12dv 5257 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
9 ofcfval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
10 ofcfval.2 . . . 4 (𝜑𝐴𝑉)
11 fnex 7254 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
129, 10, 11syl2anc 583 . . 3 (𝜑𝐹 ∈ V)
13 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
1413elexd 3512 . . 3 (𝜑𝐶 ∈ V)
159fndmd 6684 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
1615, 10eqeltrd 2844 . . . 4 (𝜑 → dom 𝐹𝑉)
1716mptexd 7261 . . 3 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
182, 8, 12, 14, 17ovmpod 7602 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
1915eleq2d 2830 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2019pm5.32i 574 . . . . 5 ((𝜑𝑥 ∈ dom 𝐹) ↔ (𝜑𝑥𝐴))
21 ofcfval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2220, 21sylbi 217 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = 𝐵)
2322oveq1d 7463 . . 3 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐹𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
2415, 23mpteq12dva 5255 . 2 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
2518, 24eqtrd 2780 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cmpt 5249  dom cdm 5700   Fn wfn 6568  cfv 6573  (class class class)co 7448  cmpo 7450  f/c cofc 34059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-ofc 34060
This theorem is referenced by:  ofcval  34063  ofcfn  34064  ofcfeqd2  34065  ofcf  34067  ofcfval2  34068  ofcc  34070  ofcof  34071
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