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Theorem ofcfval 30461
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 30459 . . . 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 811 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑓 = 𝐹)
43dmeqd 5473 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
53fveq1d 6346 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑓𝑥) = (𝐹𝑥))
6 simprr 813 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑐 = 𝐶)
75, 6oveq12d 6823 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
84, 7mpteq12dv 4877 . . 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 6637 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
129, 10, 11syl2anc 696 . . 3 (𝜑𝐹 ∈ V)
13 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
14 elex 3344 . . . 4 (𝐶𝑊𝐶 ∈ V)
1513, 14syl 17 . . 3 (𝜑𝐶 ∈ V)
16 fndm 6143 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
179, 16syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
1817, 10eqeltrd 2831 . . . 4 (𝜑 → dom 𝐹𝑉)
19 mptexg 6640 . . . 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 6945 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
2217eleq2d 2817 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2322pm5.32i 672 . . . . 5 ((𝜑𝑥 ∈ dom 𝐹) ↔ (𝜑𝑥𝐴))
24 ofcfval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2523, 24sylbi 207 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = 𝐵)
2625oveq1d 6820 . . 3 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐹𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
2717, 26mpteq12dva 4876 . 2 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
2821, 27eqtrd 2786 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  cmpt 4873  dom cdm 5258   Fn wfn 6036  cfv 6041  (class class class)co 6805  cmpt2 6807  𝑓/𝑐cofc 30458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-ofc 30459
This theorem is referenced by:  ofcval  30462  ofcfn  30463  ofcfeqd2  30464  ofcf  30466  ofcfval2  30467  ofcc  30469  ofcof  30470
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