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Theorem fnsnfv 5363
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 2090 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 5345 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2syl5bb 190 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2205 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 3452 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 9 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 imasng 4797 . . 3 (𝐵𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
87adantl 271 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
94, 6, 83eqtr4d 2130 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {cab 2074  {csn 3446   class class class wbr 3845  cima 4441   Fn wfn 5010  cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  fnimapr  5364  funfvdm  5367  fvco2  5373  fvimacnvi  5413  fsn2  5471  phplem4  6569  phplem4dom  6576  phplem4on  6581  fiintim  6637  fidcenumlemrks  6660  fidcenumlemr  6662  resunimafz0  10232
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