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Theorem funfvdm 5179
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 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5135 . . 3 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
2 unisng 3588 . . 3 ((𝐹A) V → {(𝐹A)} = (𝐹A))
31, 2syl 14 . 2 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹A))
4 eqid 2037 . . . . 5 dom 𝐹 = dom 𝐹
5 df-fn 4848 . . . . 5 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 dom 𝐹 = dom 𝐹))
64, 5mpbiran2 847 . . . 4 (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
7 fnsnfv 5175 . . . 4 ((𝐹 Fn dom 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
86, 7sylanbr 269 . . 3 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
98unieqd 3582 . 2 ((Fun 𝐹 A dom 𝐹) → {(𝐹A)} = (𝐹 “ {A}))
103, 9eqtr3d 2071 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) = (𝐹 “ {A}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367   cuni 3571  dom cdm 4288  cima 4291  Fun wfun 4839   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  funfvdm2  5180  fvun1  5182
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