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Theorem ofcf 33096
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 6718 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcf.4 . . 3 (𝜑𝐴𝑉)
4 ofcf.5 . . 3 (𝜑𝐶𝑇)
5 eqidd 2733 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
62, 3, 4, 5ofcfval 33091 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑧𝐴 ↦ ((𝐹𝑧)𝑅𝐶)))
71ffvelcdmda 7086 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
84adantr 481 . . 3 ((𝜑𝑧𝐴) → 𝐶𝑇)
9 ofcf.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
109ralrimivva 3200 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1110adantr 481 . . 3 ((𝜑𝑧𝐴) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12 ovrspc2v 7434 . . 3 ((((𝐹𝑧) ∈ 𝑆𝐶𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
137, 8, 11, 12syl21anc 836 . 2 ((𝜑𝑧𝐴) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
146, 13fmpt3d 7115 1 (𝜑 → (𝐹f/c 𝑅𝐶):𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3061  wf 6539  cfv 6543  (class class class)co 7408  f/c cofc 33088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-ofc 33089
This theorem is referenced by:  signshf  33594
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