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Theorem offun 7693
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 2728 . . 3 (𝐴𝐵) = (𝐴𝐵)
61, 2, 3, 4, 5offn 7692 . 2 (𝜑 → (𝐹f 𝑅𝐺) Fn (𝐴𝐵))
7 fnfun 6648 . 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 2099  cin 3944  Fun wfun 6536   Fn wfn 6537  (class class class)co 7414  f cof 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679
This theorem is referenced by:  lcomfsupp  20778  frlmphl  21708  frlmsslsp  21723  psrbagev1  22014  psrbagev1OLD  22015  mhpmulcl  22066  mndpsuppss  47429  mndpfsupp  47434
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