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Theorem fnsnfv 5448
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 2119 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 5430 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2syl5bb 191 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2235 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 3503 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 9 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 imasng 4874 . . 3 (𝐵𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
87adantl 275 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
94, 6, 83eqtr4d 2160 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  {cab 2103  {csn 3497   class class class wbr 3899  cima 4512   Fn wfn 5088  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fnimapr  5449  funfvdm  5452  fvco2  5458  fvimacnvi  5502  fsn2  5562  phplem4  6717  phplem4dom  6724  phplem4on  6729  fiintim  6785  fidcenumlemrks  6809  fidcenumlemr  6811  resunimafz0  10542  ennnfonelemhf1o  11853
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