Theorem funfv 6921

Description: A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.)

Assertion

Ref	Expression
funfv	⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))

Proof of Theorem funfv

Step	Hyp	Ref	Expression
1		fvex 6847	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2	1	unisn 4882	. . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴)
3		eqid 2736	. . . . . . 7 ⊢ dom 𝐹 = dom 𝐹
4		df-fn 6495	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
5	3, 4	mpbiran2 710	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6		fnsnfv 6913	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
7	5, 6	sylanbr 582	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
8	7	unieqd 4876	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴}))
9	2, 8	eqtr3id 2785	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
10	9	ex 412	. 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})))
11		ndmfv 6866	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
12		ndmima 6062	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
13	12	unieqd 4876	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅)
14		uni0 4891	. . . 4 ⊢ ∪ ∅ = ∅
15	13, 14	eqtrdi 2787	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅)
16	11, 15	eqtr4d 2774	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
17	10, 16	pm2.61d1 180	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∅c0 4285  {csn 4580  ∪ cuni 4863  dom cdm 5624   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  funfv2  6922  fvun  6924  dffv2  6929  setrecsss  49946