Theorem fnsnfv 6987

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 6101	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
2	1	adantl 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
3		velsn 4641	. . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵))
4		eqcom 2743	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦)
6		fnbrfvb 6958	. . . 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 2875	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)})
9	2, 8	eqtr2d 2777	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  {csn 4625   class class class wbr 5142   “ cima 5687   Fn wfn 6555  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568

This theorem is referenced by:  fnimapr  6991  fnimatpd  6992  funfv  6995  fvco2  7005  fvimacnvi  7071  fvimacnvALT  7076  fsn2  7155  fparlem3  8140  fparlem4  8141  suppval1  8192  suppsnop  8204  domunsncan  9113  phplem2  9246  phplem4OLD  9258  imafiOLD  9355  domunfican  9362  fiint  9367  fiintOLD  9368  infdifsn  9698  cantnfp1lem3  9721  resunimafz0  14485  symgfixelsi  19454  dprdf1o  20053  frlmlbs  21818  f1lindf  21843  cnt1  23359  xkohaus  23662  xkoptsub  23663  ustuqtop3  24253  bday1s  27877  old1  27915  madeoldsuc  27924  n0sbday  28355  zscut  28394  pw2bday  28419  zs12bday  28425  cyclnumvtx  29821  eulerpartlemmf  34378  poimirlem4  37632  poimirlem6  37634  poimirlem7  37635  poimirlem9  37637  poimirlem13  37641  poimirlem14  37642  poimirlem16  37644  poimirlem19  37647  grpokerinj  37901  fnimasnd  42272  k0004lem3  44167  funcoressn  47059  cycl3grtri  47919