Theorem chfnrn 2885

Description: The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain.

Assertion

Ref	Expression
chfnrn	⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ x) → ran F ⊆ ∪A)

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

Proof of Theorem chfnrn

Step	Hyp	Ref	Expression
1		fvelrn 2883	. . . . 5 ⊢ (F Fn A → (y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y))
2	1	biimpd 135	. . . 4 ⊢ (F Fn A → (y ∈ ran F → ∃x ∈ A (F ‘x) = y))
3		hbra1 1237	. . . . 5 ⊢ (∀x ∈ A (F ‘x) ∈ x → ∀x∀x ∈ A (F ‘x) ∈ x)
4		ra4 1243	. . . . . 6 ⊢ (∀x ∈ A (F ‘x) ∈ x → (x ∈ A → (F ‘x) ∈ x))
5		eleq1 1149	. . . . . . 7 ⊢ ((F ‘x) = y → ((F ‘x) ∈ x ↔ y ∈ x))
6	5	biimpcd 137	. . . . . 6 ⊢ ((F ‘x) ∈ x → ((F ‘x) = y → y ∈ x))
7	4, 6	syl6 23	. . . . 5 ⊢ (∀x ∈ A (F ‘x) ∈ x → (x ∈ A → ((F ‘x) = y → y ∈ x)))
8	3, 7	r19.22d 1276	. . . 4 ⊢ (∀x ∈ A (F ‘x) ∈ x → (∃x ∈ A (F ‘x) = y → ∃x ∈ A y ∈ x))
9	2, 8	sylan9 359	. . 3 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ x) → (y ∈ ran F → ∃x ∈ A y ∈ x))
10		eluni2 1923	. . 3 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
11	9, 10	syl6ibr 186	. 2 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ x) → (y ∈ ran F → y ∈ ∪A))
12	11	ssrdv 1509	1 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ x) → ran F ⊆ ∪A)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∪cuni 1919  ran crn 2411   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  ac5b 3574

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-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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