Theorem funfv 6970

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 6896	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2	1	unisn 4892	. . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴)
3		eqid 2763	. . . . . . 7 ⊢ dom 𝐹 = dom 𝐹
4		df-fn 6541	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
5	3, 4	mpbiran2 722	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6		fnsnfv 6962	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
7	5, 6	sylanbr 593	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
8	7	unieqd 4886	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴}))
9	2, 8	eqtr3id 2812	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
10	9	ex 417	. 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})))
11		ndmfv 6915	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
12		ndmima 6107	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
13	12	unieqd 4886	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅)
14		uni0 4902	. . . 4 ⊢ ∪ ∅ = ∅
15	13, 14	eqtrdi 2814	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅)
16	11, 15	eqtr4d 2801	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
17	10, 16	pm2.61d1 182	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∅c0 4287  {csn 4590  ∪ cuni 4873  dom cdm 5663   “ cima 5666  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  funfv2  6971  fvun  6973  dffv2  6978  setrecsss  50462