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Theorem fnsnfv 5175
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 2039 . . . 4 (y = (𝐹B) ↔ (𝐹B) = y)
2 fnbrfvb 5157 . . . 4 ((𝐹 Fn A B A) → ((𝐹B) = yB𝐹y))
31, 2syl5bb 181 . . 3 ((𝐹 Fn A B A) → (y = (𝐹B) ↔ B𝐹y))
43abbidv 2152 . 2 ((𝐹 Fn A B A) → {yy = (𝐹B)} = {yB𝐹y})
5 df-sn 3373 . . 3 {(𝐹B)} = {yy = (𝐹B)}
65a1i 9 . 2 ((𝐹 Fn A B A) → {(𝐹B)} = {yy = (𝐹B)})
7 imasng 4633 . . 3 (B A → (𝐹 “ {B}) = {yB𝐹y})
87adantl 262 . 2 ((𝐹 Fn A B A) → (𝐹 “ {B}) = {yB𝐹y})
94, 6, 83eqtr4d 2079 1 ((𝐹 Fn A B A) → {(𝐹B)} = (𝐹 “ {B}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  {csn 3367   class class class wbr 3755  cima 4291   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fnimapr  5176  funfvdm  5179  fvco2  5185  fvimacnvi  5224  fsn2  5280
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