Theorem ffnfv 2892

Description: A function maps to a class to which all values belong.

Assertion

Ref	Expression
ffnfv	⊢ (F:A–→B ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))

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

Proof of Theorem ffnfv

Step	Hyp	Ref	Expression
1		ffn 2752	. . 3 ⊢ (F:A–→B → F Fn A)
2		ffvrn 2890	. . . . 5 ⊢ ((F:A–→B ∧ x ∈ A) → (F ‘x) ∈ B)
3	2	exp 291	. . . 4 ⊢ (F:A–→B → (x ∈ A → (F ‘x) ∈ B))
4	3	r19.21aiv 1259	. . 3 ⊢ (F:A–→B → ∀x ∈ A (F ‘x) ∈ B)
5	1, 4	jca 236	. 2 ⊢ (F:A–→B → (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))
6		pm3.26 256	. . . 4 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → F Fn A)
7		fvelrn 2883	. . . . . . . 8 ⊢ (F Fn A → (y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y))
8	7	biimpd 135	. . . . . . 7 ⊢ (F Fn A → (y ∈ ran F → ∃x ∈ A (F ‘x) = y))
9		hbra1 1237	. . . . . . . 8 ⊢ (∀x ∈ A (F ‘x) ∈ B → ∀x∀x ∈ A (F ‘x) ∈ B)
10		ax-17 925	. . . . . . . 8 ⊢ (y ∈ B → ∀x y ∈ B)
11		ra4 1243	. . . . . . . . 9 ⊢ (∀x ∈ A (F ‘x) ∈ B → (x ∈ A → (F ‘x) ∈ B))
12		eleq1 1149	. . . . . . . . . 10 ⊢ ((F ‘x) = y → ((F ‘x) ∈ B ↔ y ∈ B))
13	12	biimpcd 137	. . . . . . . . 9 ⊢ ((F ‘x) ∈ B → ((F ‘x) = y → y ∈ B))
14	11, 13	syl6 23	. . . . . . . 8 ⊢ (∀x ∈ A (F ‘x) ∈ B → (x ∈ A → ((F ‘x) = y → y ∈ B)))
15	9, 10, 14	r19.23ad 1285	. . . . . . 7 ⊢ (∀x ∈ A (F ‘x) ∈ B → (∃x ∈ A (F ‘x) = y → y ∈ B))
16	8, 15	sylan9 359	. . . . . 6 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → (y ∈ ran F → y ∈ B))
17	16	r19.21aiv 1259	. . . . 5 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → ∀y ∈ ran Fy ∈ B)
18		dfss3 1498	. . . . 5 ⊢ (ran F ⊆ B ↔ ∀y ∈ ran Fy ∈ B)
19	17, 18	sylibr 175	. . . 4 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → ran F ⊆ B)
20	6, 19	jca 236	. . 3 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → (F Fn A ∧ ran F ⊆ B))
21		df-f 2434	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
22	20, 21	sylibr 175	. 2 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → F:A–→B)
23	5, 22	impbi 139	1 ⊢ (F:A–→B ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  fnfvrnss 2893  fopabfv 2894  abianfp 3000  ffnoprval 3041  mapxpen 3390  unblem4 3434

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-f 2434  df-fv 2438

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