MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnsnfv Structured version   Visualization version   GIF version

Theorem fnsnfv 6297
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 2658 . . . 4 (𝑦 = (𝐹𝐵) ↔ (𝐹𝐵) = 𝑦)
2 fnbrfvb 6274 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝑦𝐵𝐹𝑦))
31, 2syl5bb 272 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 = (𝐹𝐵) ↔ 𝐵𝐹𝑦))
43abbidv 2770 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {𝑦𝑦 = (𝐹𝐵)} = {𝑦𝐵𝐹𝑦})
5 df-sn 4211 . . 3 {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)}
65a1i 11 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = {𝑦𝑦 = (𝐹𝐵)})
7 fnrel 6027 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
8 relimasn 5523 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
97, 8syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
109adantr 480 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 “ {𝐵}) = {𝑦𝐵𝐹𝑦})
114, 6, 103eqtr4d 2695 1 ((𝐹 Fn 𝐴𝐵𝐴) → {(𝐹𝐵)} = (𝐹 “ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  {csn 4210   class class class wbr 4685  cima 5146  Rel wrel 5148   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fnimapr  6301  funfv  6304  fvco2  6312  fvimacnvi  6371  fvimacnvALT  6376  fsn2  6443  fparlem3  7324  fparlem4  7325  suppval1  7346  suppsnop  7354  domunsncan  8101  phplem4  8183  domunfican  8274  fiint  8278  infdifsn  8592  cantnfp1lem3  8615  resunimafz0  13267  symgfixelsi  17901  dprdf1o  18477  frlmlbs  20184  f1lindf  20209  cnt1  21202  xkohaus  21504  xkoptsub  21505  ustuqtop3  22094  eulerpartlemmf  30565  poimirlem4  33543  poimirlem6  33545  poimirlem7  33546  poimirlem9  33548  poimirlem13  33552  poimirlem14  33553  poimirlem16  33555  poimirlem19  33558  grpokerinj  33822  k0004lem3  38764  funcoressn  41528
  Copyright terms: Public domain W3C validator