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Theorem ofcfval 31256
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 31254 . . . 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 767 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑓 = 𝐹)
43dmeqd 5767 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
53fveq1d 6665 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑓𝑥) = (𝐹𝑥))
6 simprr 769 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑐 = 𝐶)
75, 6oveq12d 7163 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
84, 7mpteq12dv 5142 . . 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 6971 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
129, 10, 11syl2anc 584 . . 3 (𝜑𝐹 ∈ V)
13 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
1413elexd 3512 . . 3 (𝜑𝐶 ∈ V)
159fndmd 6449 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
1615, 10eqeltrd 2910 . . . 4 (𝜑 → dom 𝐹𝑉)
1716mptexd 6978 . . 3 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
182, 8, 12, 14, 17ovmpod 7291 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
1915eleq2d 2895 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2019pm5.32i 575 . . . . 5 ((𝜑𝑥 ∈ dom 𝐹) ↔ (𝜑𝑥𝐴))
21 ofcfval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2220, 21sylbi 218 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = 𝐵)
2322oveq1d 7160 . . 3 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐹𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
2415, 23mpteq12dva 5141 . 2 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
2518, 24eqtrd 2853 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  dom cdm 5548   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  f/c cofc 31253
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-ofc 31254
This theorem is referenced by:  ofcval  31257  ofcfn  31258  ofcfeqd2  31259  ofcf  31261  ofcfval2  31262  ofcc  31264  ofcof  31265
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