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Theorem fnsnfv 6225
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 2628 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 6203 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2syl5bb 272 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2738 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 4156 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 11 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 fnrel 5957 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
8 relimasn 5457 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
97, 8syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
109adantr 481 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
114, 6, 103eqtr4d 2665 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  {csn 4155   class class class wbr 4623  cima 5087  Rel wrel 5089   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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  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:  fnimapr  6229  funfv  6232  fvco2  6240  fvimacnvi  6297  fvimacnvALT  6302  fsn2  6368  fparlem3  7239  fparlem4  7240  suppval1  7261  suppsnop  7269  domunsncan  8020  phplem4  8102  domunfican  8193  fiint  8197  infdifsn  8514  cantnfp1lem3  8537  resunimafz0  13183  symgfixelsi  17795  dprdf1o  18371  frlmlbs  20076  f1lindf  20101  cnt1  21094  xkohaus  21396  xkoptsub  21397  ustuqtop3  21987  eulerpartlemmf  30260  poimirlem4  33084  poimirlem6  33086  poimirlem7  33087  poimirlem9  33089  poimirlem13  33093  poimirlem14  33094  poimirlem16  33096  poimirlem19  33099  grpokerinj  33363  k0004lem3  37968  funcoressn  40541
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