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Theorem ofcf 32742
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 6674 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcf.4 . . 3 (𝜑𝐴𝑉)
4 ofcf.5 . . 3 (𝜑𝐶𝑇)
5 eqidd 2738 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
62, 3, 4, 5ofcfval 32737 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑧𝐴 ↦ ((𝐹𝑧)𝑅𝐶)))
71ffvelcdmda 7040 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
84adantr 482 . . 3 ((𝜑𝑧𝐴) → 𝐶𝑇)
9 ofcf.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
109ralrimivva 3198 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1110adantr 482 . . 3 ((𝜑𝑧𝐴) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12 ovrspc2v 7388 . . 3 ((((𝐹𝑧) ∈ 𝑆𝐶𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
137, 8, 11, 12syl21anc 837 . 2 ((𝜑𝑧𝐴) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
146, 13fmpt3d 7069 1 (𝜑 → (𝐹f/c 𝑅𝐶):𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3065  wf 6497  cfv 6501  (class class class)co 7362  f/c cofc 32734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-ofc 32735
This theorem is referenced by:  signshf  33240
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