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Theorem fnsnfv 6728
 Description: Singleton of function value. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
fnsnfv ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))

Proof of Theorem fnsnfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqcom 2805 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 6703 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2syl5bb 286 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2862 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 4529 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 11 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 fnrel 6432 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
8 relimasn 5923 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
97, 8syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
109adantr 484 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
114, 6, 103eqtr4d 2843 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  {csn 4528   class class class wbr 5034   “ cima 5526  Rel wrel 5528   Fn wfn 6327  ‘cfv 6332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-fv 6340 This theorem is referenced by:  fnimapr  6732  funfv  6735  fvco2  6745  fvimacnvi  6809  fvimacnvALT  6814  fsn2  6885  fparlem3  7805  fparlem4  7806  suppval1  7832  suppsnop  7843  domunsncan  8618  phplem4  8701  domunfican  8793  fiint  8797  infdifsn  9122  cantnfp1lem3  9145  resunimafz0  13819  symgfixelsi  18576  dprdf1o  19168  frlmlbs  20508  f1lindf  20533  cnt1  21996  xkohaus  22299  xkoptsub  22300  ustuqtop3  22890  fnimatp  30484  eulerpartlemmf  31809  bday1s  33506  madeoldsuc  33545  poimirlem4  35212  poimirlem6  35214  poimirlem7  35215  poimirlem9  35217  poimirlem13  35221  poimirlem14  35222  poimirlem16  35224  poimirlem19  35227  grpokerinj  35482  k0004lem3  41023  funcoressn  43802
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