Theorem funfv 6725

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 6658	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2	1	unisn 4820	. . . 4 ⊢ ∪ {(𝐹‘𝐴)} = (𝐹‘𝐴)
3		eqid 2798	. . . . . . 7 ⊢ dom 𝐹 = dom 𝐹
4		df-fn 6327	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
5	3, 4	mpbiran2 709	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 ↔ Fun 𝐹)
6		fnsnfv 6718	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
7	5, 6	sylanbr 585	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
8	7	unieqd 4814	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∪ {(𝐹‘𝐴)} = ∪ (𝐹 “ {𝐴}))
9	2, 8	syl5eqr 2847	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
10	9	ex 416	. 2 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})))
11		ndmfv 6675	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
12		ndmima 5933	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = ∅)
13	12	unieqd 4814	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ ∅)
14		uni0 4828	. . . 4 ⊢ ∪ ∅ = ∅
15	13, 14	eqtrdi 2849	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ∪ (𝐹 “ {𝐴}) = ∅)
16	11, 15	eqtr4d 2836	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
17	10, 16	pm2.61d1 183	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4243  {csn 4525  ∪ cuni 4800  dom cdm 5519   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332

This theorem is referenced by:  funfv2  6726  fvun  6728  dffv2  6733  setrecsss  45230