Theorem fnsnfv 6168

Description: Singleton of function value. (Contributed by NM, 22-May-1998.)

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		eqcom 2617	. . . 4 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦)
2		fnbrfvb 6146	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦))
3	1, 2	syl5bb 271	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑦 = (𝐹‘𝐵) ↔ 𝐵𝐹𝑦))
4	3	abbidv 2728	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} = {𝑦 ∣ 𝐵𝐹𝑦})
5		df-sn 4126	. . 3 ⊢ {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)}
6	5	a1i 11	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)})
7		fnrel 5903	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
8		relimasn 5407	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
9	7, 8	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
10	9	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
11	4, 6, 10	3eqtr4d 2654	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {csn 4125   class class class wbr 4583   “ cima 5041  Rel wrel 5043   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by:  fnimapr  6172  funfv  6175  fvco2  6183  fvimacnvi  6239  fvimacnvALT  6244  fsn2  6309  fparlem3  7166  fparlem4  7167  suppval1  7188  suppsnop  7196  domunsncan  7945  phplem4  8027  domunfican  8118  fiint  8122  infdifsn  8437  cantnfp1lem3  8460  symgfixelsi  17678  dprdf1o  18254  frlmlbs  19955  f1lindf  19980  cnt1  20964  xkohaus  21266  xkoptsub  21267  ustuqtop3  21857  2pthlem2  26126  eupath2lem3  26506  eulerpartlemmf  29764  poimirlem4  32583  poimirlem6  32585  poimirlem7  32586  poimirlem9  32588  poimirlem13  32592  poimirlem14  32593  poimirlem16  32595  poimirlem19  32598  grpokerinj  32862  k0004lem3  37467  funcoressn  39856  resunimafz0  40368