Theorem fnsnfv 6481

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 2804	. . . 4 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦)
2		fnbrfvb 6458	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦))
3	1, 2	syl5bb 275	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑦 = (𝐹‘𝐵) ↔ 𝐵𝐹𝑦))
4	3	abbidv 2916	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} = {𝑦 ∣ 𝐵𝐹𝑦})
5		df-sn 4367	. . 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 6198	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
8		relimasn 5703	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
9	7, 8	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
10	9	adantr 473	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦})
11	4, 6, 10	3eqtr4d 2841	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {cab 2783  {csn 4366   class class class wbr 4841   “ cima 5313  Rel wrel 5315   Fn wfn 6094  ‘cfv 6099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107

This theorem is referenced by:  fnimapr  6485  funfv  6488  fvco2  6496  fvimacnvi  6555  fvimacnvALT  6560  fsn2  6628  fparlem3  7514  fparlem4  7515  suppval1  7536  suppsnop  7544  domunsncan  8300  phplem4  8382  domunfican  8473  fiint  8477  infdifsn  8802  cantnfp1lem3  8825  resunimafz0  13474  symgfixelsi  18164  dprdf1o  18744  frlmlbs  20458  f1lindf  20483  cnt1  21480  xkohaus  21782  xkoptsub  21783  ustuqtop3  22372  eulerpartlemmf  30945  poimirlem4  33894  poimirlem6  33896  poimirlem7  33897  poimirlem9  33899  poimirlem13  33903  poimirlem14  33904  poimirlem16  33906  poimirlem19  33909  grpokerinj  34171  k0004lem3  39217  funcoressn  41913