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Theorem ofcf 30505
 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)
21ffnd 6185 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcf.4 . . 3 (𝜑𝐴𝑉)
4 ofcf.5 . . 3 (𝜑𝐶𝑇)
5 eqidd 2772 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) = (𝐹𝑧))
62, 3, 4, 5ofcfval 30500 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑧𝐴 ↦ ((𝐹𝑧)𝑅𝐶)))
71ffvelrnda 6504 . . 3 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ 𝑆)
84adantr 466 . . 3 ((𝜑𝑧𝐴) → 𝐶𝑇)
9 ofcf.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
109ralrimivva 3120 . . . 4 (𝜑 → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
1110adantr 466 . . 3 ((𝜑𝑧𝐴) → ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
12 ovrspc2v 6821 . . 3 ((((𝐹𝑧) ∈ 𝑆𝐶𝑇) ∧ ∀𝑥𝑆𝑦𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
137, 8, 11, 12syl21anc 1475 . 2 ((𝜑𝑧𝐴) → ((𝐹𝑧)𝑅𝐶) ∈ 𝑈)
146, 13fmpt3d 6531 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶):𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 2145  ∀wral 3061  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796  ∘𝑓/𝑐cofc 30497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-ofc 30498 This theorem is referenced by:  signshf  31005
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