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Theorem fnsnfv 5157
Description: Singleton of function value. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
fnsnfv ((𝐹 Fn A B A) → {(𝐹B)} = (𝐹 “ {B}))

Proof of Theorem fnsnfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqcom 2024 . . . 4 (y = (𝐹B) ↔ (𝐹B) = y)
2 fnbrfvb 5139 . . . 4 ((𝐹 Fn A B A) → ((𝐹B) = yB𝐹y))
31, 2syl5bb 181 . . 3 ((𝐹 Fn A B A) → (y = (𝐹B) ↔ B𝐹y))
43abbidv 2137 . 2 ((𝐹 Fn A B A) → {yy = (𝐹B)} = {yB𝐹y})
5 df-sn 3356 . . 3 {(𝐹B)} = {yy = (𝐹B)}
65a1i 9 . 2 ((𝐹 Fn A B A) → {(𝐹B)} = {yy = (𝐹B)})
7 imasng 4617 . . 3 (B A → (𝐹 “ {B}) = {yB𝐹y})
87adantl 262 . 2 ((𝐹 Fn A B A) → (𝐹 “ {B}) = {yB𝐹y})
94, 6, 83eqtr4d 2064 1 ((𝐹 Fn A B A) → {(𝐹B)} = (𝐹 “ {B}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  {cab 2008  {csn 3350   class class class wbr 3738  cima 4275   Fn wfn 4824  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  fnimapr  5158  funfvdm  5161  fvco2  5167  fvimacnvi  5206  fsn2  5262
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