Theorem fnsnfv 6988

Description: Singleton of function value. (Contributed by NM, 22-May-1998.) (Proof shortened by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
fnsnfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Proof of Theorem fnsnfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasng 6104	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
2	1	adantl 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
3		velsn 4647	. . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵))
4		eqcom 2742	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦)
6		fnbrfvb 6960	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦))
7	5, 6	bitr2id 284	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝑦 ↔ 𝑦 ∈ {(𝐹‘𝐵)}))
8	7	eqabcdv 2874	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)})
9	2, 8	eqtr2d 2776	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  {csn 4631   class class class wbr 5148   “ cima 5692   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571

This theorem is referenced by:  fnimapr  6992  fnimatpd  6993  funfv  6996  fvco2  7006  fvimacnvi  7072  fvimacnvALT  7077  fsn2  7156  fparlem3  8138  fparlem4  8139  suppval1  8190  suppsnop  8202  domunsncan  9111  phplem2  9243  phplem4OLD  9255  imafiOLD  9352  domunfican  9359  fiint  9364  fiintOLD  9365  infdifsn  9695  cantnfp1lem3  9718  resunimafz0  14481  symgfixelsi  19468  dprdf1o  20067  frlmlbs  21835  f1lindf  21860  cnt1  23374  xkohaus  23677  xkoptsub  23678  ustuqtop3  24268  bday1s  27891  old1  27929  madeoldsuc  27938  n0sbday  28369  zscut  28408  pw2bday  28433  zs12bday  28439  eulerpartlemmf  34357  poimirlem4  37611  poimirlem6  37613  poimirlem7  37614  poimirlem9  37616  poimirlem13  37620  poimirlem14  37621  poimirlem16  37623  poimirlem19  37626  grpokerinj  37880  fnimasnd  42251  k0004lem3  44139  funcoressn  46992