Theorem fvelrnb 5164

Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Assertion

Ref	Expression
fvelrnb	⊢ (𝐹 Fn A → (B ∈ ran 𝐹 ↔ ∃x ∈ A (𝐹‘x) = B))

Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem fvelrnb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2306	. . . 4 ⊢ (∃x ∈ A (𝐹‘x) = B ↔ ∃x(x ∈ A ∧ (𝐹‘x) = B))
2		19.41v 1779	. . . . 5 ⊢ (∃x((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) ↔ (∃x(x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A))
3		simpl 102	. . . . . . . . . 10 ⊢ ((x ∈ A ∧ (𝐹‘x) = B) → x ∈ A)
4	3	anim1i 323	. . . . . . . . 9 ⊢ (((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → (x ∈ A ∧ 𝐹 Fn A))
5	4	ancomd 254	. . . . . . . 8 ⊢ (((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → (𝐹 Fn A ∧ x ∈ A))
6		funfvex 5135	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
7	6	funfni 4942	. . . . . . . 8 ⊢ ((𝐹 Fn A ∧ x ∈ A) → (𝐹‘x) ∈ V)
8	5, 7	syl 14	. . . . . . 7 ⊢ (((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → (𝐹‘x) ∈ V)
9		simpr 103	. . . . . . . . 9 ⊢ ((x ∈ A ∧ (𝐹‘x) = B) → (𝐹‘x) = B)
10	9	eleq1d 2103	. . . . . . . 8 ⊢ ((x ∈ A ∧ (𝐹‘x) = B) → ((𝐹‘x) ∈ V ↔ B ∈ V))
11	10	adantr 261	. . . . . . 7 ⊢ (((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → ((𝐹‘x) ∈ V ↔ B ∈ V))
12	8, 11	mpbid 135	. . . . . 6 ⊢ (((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → B ∈ V)
13	12	exlimiv 1486	. . . . 5 ⊢ (∃x((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → B ∈ V)
14	2, 13	sylbir 125	. . . 4 ⊢ ((∃x(x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) → B ∈ V)
15	1, 14	sylanb 268	. . 3 ⊢ ((∃x ∈ A (𝐹‘x) = B ∧ 𝐹 Fn A) → B ∈ V)
16	15	expcom 109	. 2 ⊢ (𝐹 Fn A → (∃x ∈ A (𝐹‘x) = B → B ∈ V))
17		fnrnfv 5163	. . . 4 ⊢ (𝐹 Fn A → ran 𝐹 = {y ∣ ∃x ∈ A y = (𝐹‘x)})
18	17	eleq2d 2104	. . 3 ⊢ (𝐹 Fn A → (B ∈ ran 𝐹 ↔ B ∈ {y ∣ ∃x ∈ A y = (𝐹‘x)}))
19		eqeq1 2043	. . . . . 6 ⊢ (y = B → (y = (𝐹‘x) ↔ B = (𝐹‘x)))
20		eqcom 2039	. . . . . 6 ⊢ (B = (𝐹‘x) ↔ (𝐹‘x) = B)
21	19, 20	syl6bb 185	. . . . 5 ⊢ (y = B → (y = (𝐹‘x) ↔ (𝐹‘x) = B))
22	21	rexbidv 2321	. . . 4 ⊢ (y = B → (∃x ∈ A y = (𝐹‘x) ↔ ∃x ∈ A (𝐹‘x) = B))
23	22	elab3g 2687	. . 3 ⊢ ((∃x ∈ A (𝐹‘x) = B → B ∈ V) → (B ∈ {y ∣ ∃x ∈ A y = (𝐹‘x)} ↔ ∃x ∈ A (𝐹‘x) = B))
24	18, 23	sylan9bbr 436	. 2 ⊢ (((∃x ∈ A (𝐹‘x) = B → B ∈ V) ∧ 𝐹 Fn A) → (B ∈ ran 𝐹 ↔ ∃x ∈ A (𝐹‘x) = B))
25	16, 24	mpancom 399	1 ⊢ (𝐹 Fn A → (B ∈ ran 𝐹 ↔ ∃x ∈ A (𝐹‘x) = B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  ran crn 4289   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  chfnrn  5221  rexrn  5247  ralrn  5248  elrnrexdmb  5250  ffnfv  5266  fconstfvm  5322  elunirn  5348  isoini  5400  reldm  5754  uzn0  8224  frec2uzrand  8832  frecuzrdgfn  8839