Theorem funfv 2862

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 2861	. . . . . 6 ⊢ ((F Fn dom F ∧ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
2		df-fn 2433	. . . . . . 7 ⊢ (F Fn dom F ↔ (Fun F ∧ dom F = dom F))
3		cleqid 1102	. . . . . . 7 ⊢ dom F = dom F
4	2, 3	mpbiranr 548	. . . . . 6 ⊢ (F Fn dom F ↔ Fun F)
5	1, 4	sylanbr 345	. . . . 5 ⊢ ((Fun F ∧ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
6	5	unieqd 1929	. . . 4 ⊢ ((Fun F ∧ A ∈ dom F) → ∪{(F ‘A)} = ∪(F “ {A}))
7		fvex 2838	. . . . 5 ⊢ (F ‘A) ∈ V
8	7	unisn 1932	. . . 4 ⊢ ∪{(F ‘A)} = (F ‘A)
9	6, 8	syl5eqr 1138	. . 3 ⊢ ((Fun F ∧ A ∈ dom F) → (F ‘A) = ∪(F “ {A}))
10	9	exp 291	. 2 ⊢ (Fun F → (A ∈ dom F → (F ‘A) = ∪(F “ {A})))
11		ndmfv 2848	. . . 4 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
12		ndmima 2623	. . . . . 6 ⊢ (¬ A ∈ dom F → (F “ {A}) = ∅)
13	12	unieqd 1929	. . . . 5 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∪∅)
14		uni0 1938	. . . . 5 ⊢ ∪∅ = ∅
15	13, 14	syl6eq 1140	. . . 4 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∅)
16	11, 15	eqtr4d 1131	. . 3 ⊢ (¬ A ∈ dom F → (F ‘A) = ∪(F “ {A}))
17	16	a1i 7	. 2 ⊢ (Fun F → (¬ A ∈ dom F → (F ‘A) = ∪(F “ {A})))
18	10, 17	pm2.61d 112	1 ⊢ (Fun F → (F ‘A) = ∪(F “ {A}))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {csn 1808  ∪cuni 1919  dom cdm 2410   “ cima 2413  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  funfv2 2863

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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