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Theorem ofcf 29492
 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/𝑐𝑅𝐶):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)
2 ffn 5958 . . . 4 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
31, 2syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
4 ofcf.4 . . 3 (𝜑𝐴𝑉)
5 ofcf.5 . . 3 (𝜑𝐶𝑇)
6 eqidd 2611 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
73, 4, 5, 6ofcfval 29487 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑧𝐴 ↦ ((𝐹𝑧)𝑅𝐶)))
81ffvelrnda 6267 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
95adantr 480 . . 3 ((𝜑𝑧𝐴) → 𝐶𝑇)
10 ofcf.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
1110ralrimivva 2954 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1211adantr 480 . . 3 ((𝜑𝑧𝐴) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
13 ovrspc2v 6571 . . 3 ((((𝐹𝑧) ∈ 𝑆𝐶𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
148, 9, 12, 13syl21anc 1317 . 2 ((𝜑𝑧𝐴) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
157, 14fmpt3d 6293 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶):𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ∘𝑓/𝑐cofc 29484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ofc 29485 This theorem is referenced by:  signshf  29991
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