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Theorem offun 7632
Description: The function operation produces a function. (Contributed by SN, 23-Jul-2024.)
Hypotheses
Ref Expression
offun.1 (𝜑𝐹 Fn 𝐴)
offun.2 (𝜑𝐺 Fn 𝐵)
offun.3 (𝜑𝐴𝑉)
offun.4 (𝜑𝐵𝑊)
Assertion
Ref Expression
offun (𝜑 → Fun (𝐹f 𝑅𝐺))

Proof of Theorem offun
StepHypRef Expression
1 offun.1 . . 3 (𝜑𝐹 Fn 𝐴)
2 offun.2 . . 3 (𝜑𝐺 Fn 𝐵)
3 offun.3 . . 3 (𝜑𝐴𝑉)
4 offun.4 . . 3 (𝜑𝐵𝑊)
5 eqid 2737 . . 3 (𝐴𝐵) = (𝐴𝐵)
61, 2, 3, 4, 5offn 7631 . 2 (𝜑 → (𝐹f 𝑅𝐺) Fn (𝐴𝐵))
7 fnfun 6603 . 2 ((𝐹f 𝑅𝐺) Fn (𝐴𝐵) → Fun (𝐹f 𝑅𝐺))
86, 7syl 17 1 (𝜑 → Fun (𝐹f 𝑅𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cin 3910  Fun wfun 6491   Fn wfn 6492  (class class class)co 7358  f cof 7616
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 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618
This theorem is referenced by:  lcomfsupp  20365  frlmsslsp  21205  psrbagev1  21488  psrbagev1OLD  21489  mhpmulcl  21542  mndpsuppss  46454  mndpfsupp  46459
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