Theorem fniunfv 2860

Description: The indexed union of a function's values is the union of its range.

Assertion

Ref	Expression
fniunfv	⊢ (F Fn A → ∪x ∈ A (F ‘x) = ∪ran F)

Distinct variable group(s):   x,A   x,F

Proof of Theorem fniunfv

Step	Hyp	Ref	Expression
1		fndm 2723	. . . . . . . . . . 11 ⊢ (F Fn A → dom F = A)
2	1	eleq2d 1156	. . . . . . . . . 10 ⊢ (F Fn A → (x ∈ dom F ↔ x ∈ A))
3		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
4	3	opeldm 2534	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ F → x ∈ dom F)
5	2, 4	syl5bi 183	. . . . . . . . 9 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
6		pm4.71r 482	. . . . . . . . 9 ⊢ ((⟨x, y⟩ ∈ F → x ∈ A) ↔ (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
7	5, 6	sylib 173	. . . . . . . 8 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
8		visset 1350	. . . . . . . . . . . 12 ⊢ y ∈ V
9	8	fnfvop 2856	. . . . . . . . . . 11 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
10		cleqcom 1103	. . . . . . . . . . 11 ⊢ ((F ‘x) = y ↔ y = (F ‘x))
11	9, 10	syl5rbbr 413	. . . . . . . . . 10 ⊢ ((F Fn A ∧ x ∈ A) → (⟨x, y⟩ ∈ F ↔ y = (F ‘x)))
12	11	exp 291	. . . . . . . . 9 ⊢ (F Fn A → (x ∈ A → (⟨x, y⟩ ∈ F ↔ y = (F ‘x))))
13	12	pm5.32d 491	. . . . . . . 8 ⊢ (F Fn A → ((x ∈ A ∧ ⟨x, y⟩ ∈ F) ↔ (x ∈ A ∧ y = (F ‘x))))
14	7, 13	bitrd 406	. . . . . . 7 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ y = (F ‘x))))
15	14	biexdv 936	. . . . . 6 ⊢ (F Fn A → (∃x⟨x, y⟩ ∈ F ↔ ∃x(x ∈ A ∧ y = (F ‘x))))
16		df-rex 1206	. . . . . 6 ⊢ (∃x ∈ A y = (F ‘x) ↔ ∃x(x ∈ A ∧ y = (F ‘x)))
17	15, 16	syl6bbr 416	. . . . 5 ⊢ (F Fn A → (∃x⟨x, y⟩ ∈ F ↔ ∃x ∈ A y = (F ‘x)))
18	17	biabdv 1183	. . . 4 ⊢ (F Fn A → {y∣∃x⟨x, y⟩ ∈ F} = {y∣∃x ∈ A y = (F ‘x)})
19		dfrn3 2524	. . . 4 ⊢ ran F = {y∣∃x⟨x, y⟩ ∈ F}
20	18, 19	syl5req 1137	. . 3 ⊢ (F Fn A → {y∣∃x ∈ A y = (F ‘x)} = ran F)
21	20	unieqd 1929	. 2 ⊢ (F Fn A → ∪{y∣∃x ∈ A y = (F ‘x)} = ∪ran F)
22		fvex 2838	. . 3 ⊢ (F ‘x) ∈ V
23	22	dfiun2 2014	. 2 ⊢ ∪x ∈ A (F ‘x) = ∪{y∣∃x ∈ A y = (F ‘x)}
24	21, 23	syl5eq 1136	1 ⊢ (F Fn A → ∪x ∈ A (F ‘x) = ∪ran F)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810  ∪cuni 1919  ∪ciun 1994  dom cdm 2410  ran crn 2411   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  unir1 3511

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-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-iun 1996  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  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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