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Theorem ofcfval 30766
 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/𝑐𝑅𝐶) = (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 30764 . . . 4 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
21a1i 11 . . 3 (𝜑 → ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))))
3 simprl 761 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑓 = 𝐹)
43dmeqd 5573 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
53fveq1d 6450 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑓𝑥) = (𝐹𝑥))
6 simprr 763 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑐 = 𝐶)
75, 6oveq12d 6942 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
84, 7mpteq12dv 4971 . . 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 6755 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
129, 10, 11syl2anc 579 . . 3 (𝜑𝐹 ∈ V)
13 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
14 elex 3414 . . . 4 (𝐶𝑊𝐶 ∈ V)
1513, 14syl 17 . . 3 (𝜑𝐶 ∈ V)
16 fndm 6237 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
179, 16syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
1817, 10eqeltrd 2859 . . . 4 (𝜑 → dom 𝐹𝑉)
19 mptexg 6758 . . . 4 (dom 𝐹𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
2018, 19syl 17 . . 3 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
212, 8, 12, 15, 20ovmpt2d 7067 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
2217eleq2d 2845 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2322pm5.32i 570 . . . . 5 ((𝜑𝑥 ∈ dom 𝐹) ↔ (𝜑𝑥𝐴))
24 ofcfval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2523, 24sylbi 209 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = 𝐵)
2625oveq1d 6939 . . 3 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐹𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
2717, 26mpteq12dva 4970 . 2 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
2821, 27eqtrd 2814 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ↦ cmpt 4967  dom cdm 5357   Fn wfn 6132  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  ∘𝑓/𝑐cofc 30763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-ofc 30764 This theorem is referenced by:  ofcval  30767  ofcfn  30768  ofcfeqd2  30769  ofcf  30771  ofcfval2  30772  ofcc  30774  ofcof  30775
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