Theorem funfv 3776

Description: A simplified expression for the value of a function when we know it's a function.

Assertion

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

Proof of Theorem funfv

Step	Hyp	Ref	Expression
1		fnsnfv 3773	. . . . . 6 ⊢ ((F Fn dom F ⋀ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
2		df-fn 3199	. . . . . . 7 ⊢ (F Fn dom F ↔ (Fun F ⋀ dom F = dom F))
3		eqid 1478	. . . . . . 7 ⊢ dom F = dom F
4	2, 3	mpbiran2 731	. . . . . 6 ⊢ (F Fn dom F ↔ Fun F)
5	1, 4	sylanbr 452	. . . . 5 ⊢ ((Fun F ⋀ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
6	5	unieqd 2516	. . . 4 ⊢ ((Fun F ⋀ A ∈ dom F) → ∪{(F ‘A)} = ∪(F “ {A}))
7		fvex 3738	. . . . 5 ⊢ (F ‘A) ∈ V
8	7	unisn 2521	. . . 4 ⊢ ∪{(F ‘A)} = (F ‘A)
9	6, 8	syl5eqr 1524	. . 3 ⊢ ((Fun F ⋀ A ∈ dom F) → (F ‘A) = ∪(F “ {A}))
10	9	ex 373	. 2 ⊢ (Fun F → (A ∈ dom F → (F ‘A) = ∪(F “ {A})))
11		ndmfv 3751	. . 3 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
12		ndmima 3440	. . . . 5 ⊢ (¬ A ∈ dom F → (F “ {A}) = ∅)
13	12	unieqd 2516	. . . 4 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∪∅)
14		uni0 2529	. . . 4 ⊢ ∪∅ = ∅
15	13, 14	syl6eq 1526	. . 3 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∅)
16	11, 15	eqtr4d 1513	. 2 ⊢ (¬ A ∈ dom F → (F ‘A) = ∪(F “ {A}))
17	10, 16	pm2.61d1 128	1 ⊢ (Fun F → (F ‘A) = ∪(F “ {A}))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∅c0 2283  {csn 2413  ∪cuni 2507  dom cdm 3176   “ cima 3179  Fun wfun 3182   Fn wfn 3183   ‘cfv 3188

This theorem is referenced by:  funfv2 3777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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