Theorem funfv 6966

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 6892	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2	1	unisn 4892	. . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴)
3		eqid 2769	. . . . . . 7 ⊢ dom 𝐹 = dom 𝐹
4		df-fn 6536	. . . . . . 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 6958	. . . . . 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 2818	. . 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 6911	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
12		ndmima 6103	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
13	12	unieqd 4886	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅)
14		uni0 4902	. . . 4 ⊢ ∪ ∅ = ∅
15	13, 14	eqtrdi 2820	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅)
16	11, 15	eqtr4d 2807	. 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 1567   ∈ wcel 2149  ∅c0 4294  {csn 4591  ∪ cuni 4873  dom cdm 5659   “ cima 5662  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is referenced by:  funfv2  6967  fvun  6969  dffv2  6974  setrecsss  50357