Theorem fnsnfv 6922

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 6044	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
2	1	adantl 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
3		velsn 4601	. . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵))
4		eqcom 2736	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦)
6		fnbrfvb 6893	. . . 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 2862	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)})
9	2, 8	eqtr2d 2765	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {csn 4585   class class class wbr 5102   “ cima 5634   Fn wfn 6494  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  fnimapr  6926  fnimatpd  6927  funfv  6930  fvco2  6940  fvimacnvi  7006  fvimacnvALT  7011  fsn2  7090  fnimasnd  7322  fparlem3  8070  fparlem4  8071  suppval1  8122  suppsnop  8134  domunsncan  9018  phplem2  9146  imafiOLD  9241  domunfican  9248  fiint  9253  fiintOLD  9254  infdifsn  9588  cantnfp1lem3  9611  resunimafz0  14388  symgfixelsi  19350  dprdf1o  19949  frlmlbs  21740  f1lindf  21765  cnt1  23271  xkohaus  23574  xkoptsub  23575  ustuqtop3  24165  bday1s  27781  old1  27825  madeoldsuc  27835  n0sbday  28285  zscut  28336  zs12bday  28397  cyclnumvtx  29781  eulerpartlemmf  34360  poimirlem4  37612  poimirlem6  37614  poimirlem7  37615  poimirlem9  37617  poimirlem13  37621  poimirlem14  37622  poimirlem16  37624  poimirlem19  37627  grpokerinj  37881  k0004lem3  44132  funcoressn  47037  cycl3grtri  47940  imaf1homlem  49090