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Theorem fnsnfv 5596
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 2191 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 5577 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2bitrid 192 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2307 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 3613 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 9 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 imasng 5011 . . 3 (𝐵𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
87adantl 277 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
94, 6, 83eqtr4d 2232 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  {cab 2175  {csn 3607   class class class wbr 4018  cima 4647   Fn wfn 5230  cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  fnimapr  5597  funfvdm  5600  fvco2  5606  fvimacnvi  5651  fsn2  5711  phplem4  6884  phplem4dom  6891  phplem4on  6896  fiintim  6958  fidcenumlemrks  6983  fidcenumlemr  6985  resunimafz0  10846  ennnfonelemhf1o  12467
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