Theorem fvrn 2888

Description: A function's value belongs to its range.

Assertion

Ref	Expression
fvrn	⊢ ((Fun F ∧ A ∈ dom F) → (F ‘A) ∈ ran F)

Proof of Theorem fvrn

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ dom F ↔ A ∈ dom F))
2	1	anbi2d 468	. . . 4 ⊢ (x = A → ((Fun F ∧ x ∈ dom F) ↔ (Fun F ∧ A ∈ dom F)))
3		fveq2 2832	. . . . 5 ⊢ (x = A → (F ‘x) = (F ‘A))
4	3	eleq1d 1155	. . . 4 ⊢ (x = A → ((F ‘x) ∈ ran F ↔ (F ‘A) ∈ ran F))
5	2, 4	imbi12d 474	. . 3 ⊢ (x = A → (((Fun F ∧ x ∈ dom F) → (F ‘x) ∈ ran F) ↔ ((Fun F ∧ A ∈ dom F) → (F ‘A) ∈ ran F)))
6		funopfv 2886	. . . . 5 ⊢ ((Fun F ∧ x ∈ dom F) → ⟨x, (F ‘x)⟩ ∈ F)
7		visset 1350	. . . . . 6 ⊢ x ∈ V
8		opeq1 1876	. . . . . . 7 ⊢ (y = x → ⟨y, (F ‘x)⟩ = ⟨x, (F ‘x)⟩)
9	8	eleq1d 1155	. . . . . 6 ⊢ (y = x → (⟨y, (F ‘x)⟩ ∈ F ↔ ⟨x, (F ‘x)⟩ ∈ F))
10	7, 9	cla4ev 1401	. . . . 5 ⊢ (⟨x, (F ‘x)⟩ ∈ F → ∃y⟨y, (F ‘x)⟩ ∈ F)
11	6, 10	syl 12	. . . 4 ⊢ ((Fun F ∧ x ∈ dom F) → ∃y⟨y, (F ‘x)⟩ ∈ F)
12		fvex 2838	. . . . 5 ⊢ (F ‘x) ∈ V
13	12	elrn 2562	. . . 4 ⊢ ((F ‘x) ∈ ran F ↔ ∃y⟨y, (F ‘x)⟩ ∈ F)
14	11, 13	sylibr 175	. . 3 ⊢ ((Fun F ∧ x ∈ dom F) → (F ‘x) ∈ ran F)
15	5, 14	vtoclg 1383	. 2 ⊢ (A ∈ dom F → ((Fun F ∧ A ∈ dom F) → (F ‘A) ∈ ran F))
16	15	anabsi7 379	1 ⊢ ((Fun F ∧ A ∈ dom F) → (F ‘A) ∈ ran F)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  dom cdm 2410  ran crn 2411  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  fnfvrn 2889  funfvima 2904  tz7.48-2 2995

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-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

metamath.org