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Theorem funfvdm 5575
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5528 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
2 unisng 3824 . . 3 ((𝐹𝐴) ∈ V → {(𝐹𝐴)} = (𝐹𝐴))
31, 2syl 14 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹𝐴))
4 eqid 2177 . . . . 5 dom 𝐹 = dom 𝐹
5 df-fn 5215 . . . . 5 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
64, 5mpbiran2 941 . . . 4 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7 fnsnfv 5571 . . . 4 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
86, 7sylanbr 285 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
98unieqd 3818 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
103, 9eqtr3d 2212 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  {csn 3591   cuni 3807  dom cdm 4623  cima 4626  Fun wfun 5206   Fn wfn 5207  cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220
This theorem is referenced by:  funfvdm2  5576  fvun1  5578
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