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Theorem ofcf 33593
Description: The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
ofcf.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
ofcf.2 (𝜑𝐹:𝐴𝑆)
ofcf.4 (𝜑𝐴𝑉)
ofcf.5 (𝜑𝐶𝑇)
Assertion
Ref Expression
ofcf (𝜑 → (𝐹f/c 𝑅𝐶):𝐴𝑈)
Distinct variable groups:   𝑦,𝐶   𝑥,𝑦,𝐹   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem ofcf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ofcf.2 . . . 4 (𝜑𝐹:𝐴𝑆)
21ffnd 6709 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcf.4 . . 3 (𝜑𝐴𝑉)
4 ofcf.5 . . 3 (𝜑𝐶𝑇)
5 eqidd 2725 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
62, 3, 4, 5ofcfval 33588 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑧𝐴 ↦ ((𝐹𝑧)𝑅𝐶)))
71ffvelcdmda 7077 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
84adantr 480 . . 3 ((𝜑𝑧𝐴) → 𝐶𝑇)
9 ofcf.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
109ralrimivva 3192 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1110adantr 480 . . 3 ((𝜑𝑧𝐴) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12 ovrspc2v 7428 . . 3 ((((𝐹𝑧) ∈ 𝑆𝐶𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
137, 8, 11, 12syl21anc 835 . 2 ((𝜑𝑧𝐴) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
146, 13fmpt3d 7108 1 (𝜑 → (𝐹f/c 𝑅𝐶):𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3053  wf 6530  cfv 6534  (class class class)co 7402  f/c cofc 33585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-ofc 33586
This theorem is referenced by:  signshf  34091
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