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Theorem funfv 6232
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
funfv (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))

Proof of Theorem funfv
StepHypRef Expression
1 fvex 6168 . . . . 5 (𝐹𝐴) ∈ V
21unisn 4424 . . . 4 {(𝐹𝐴)} = (𝐹𝐴)
3 eqid 2621 . . . . . . 7 dom 𝐹 = dom 𝐹
4 df-fn 5860 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
53, 4mpbiran2 953 . . . . . 6 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6 fnsnfv 6225 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
75, 6sylanbr 490 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
87unieqd 4419 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
92, 8syl5eqr 2669 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹 “ {𝐴}))
109ex 450 . 2 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴})))
11 ndmfv 6185 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
12 ndmima 5471 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
1312unieqd 4419 . . . 4 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
14 uni0 4438 . . . 4 ∅ = ∅
1513, 14syl6eq 2671 . . 3 𝐴 ∈ dom 𝐹 (𝐹 “ {𝐴}) = ∅)
1611, 15eqtr4d 2658 . 2 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
1710, 16pm2.61d1 171 1 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3897  {csn 4155   cuni 4409  dom cdm 5084  cima 5087  Fun wfun 5851   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  funfv2  6233  fvun  6235  dffv2  6238
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