Theorem fvelrnb 5146

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 2290	. . . 4 ⊢ (∃x ∈ A (𝐹‘x) = B ↔ ∃x(x ∈ A ∧ (𝐹‘x) = B))
2		19.41v 1764	. . . . 5 ⊢ (∃x((x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A) ↔ (∃x(x ∈ A ∧ (𝐹‘x) = B) ∧ 𝐹 Fn A))
3		ax-ia1 99	. . . . . . . . . 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 5117	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
7	6	funfni 4925	. . . . . . . 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		ax-ia2 100	. . . . . . . . 9 ⊢ ((x ∈ A ∧ (𝐹‘x) = B) → (𝐹‘x) = B)
10	9	eleq1d 2088	. . . . . . . 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 1471	. . . . 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 5145	. . . 4 ⊢ (𝐹 Fn A → ran 𝐹 = {y ∣ ∃x ∈ A y = (𝐹‘x)})
18	17	eleq2d 2089	. . 3 ⊢ (𝐹 Fn A → (B ∈ ran 𝐹 ↔ B ∈ {y ∣ ∃x ∈ A y = (𝐹‘x)}))
19		eqeq1 2028	. . . . . 6 ⊢ (y = B → (y = (𝐹‘x) ↔ B = (𝐹‘x)))
20		eqcom 2024	. . . . . 6 ⊢ (B = (𝐹‘x) ↔ (𝐹‘x) = B)
21	19, 20	syl6bb 185	. . . . 5 ⊢ (y = B → (y = (𝐹‘x) ↔ (𝐹‘x) = B))
22	21	rexbidv 2305	. . . 4 ⊢ (y = B → (∃x ∈ A y = (𝐹‘x) ↔ ∃x ∈ A (𝐹‘x) = B))
23	22	elab3g 2670	. . 3 ⊢ ((∃x ∈ A (𝐹‘x) = B → B ∈ V) → (B ∈ {y ∣ ∃x ∈ A y = (𝐹‘x)} ↔ ∃x ∈ A (𝐹‘x) = B))
24	18, 23	sylan9bbr 439	. 2 ⊢ (((∃x ∈ A (𝐹‘x) = B → B ∈ V) ∧ 𝐹 Fn A) → (B ∈ ran 𝐹 ↔ ∃x ∈ A (𝐹‘x) = B))
25	16, 24	mpancom 401	1 ⊢ (𝐹 Fn A → (B ∈ ran 𝐹 ↔ ∃x ∈ A (𝐹‘x) = B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  ∃wrex 2285  Vcvv 2535  ran crn 4273   Fn wfn 4824  ‘cfv 4829

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by:  chfnrn  5203  rexrn  5229  ralrn  5230  elrnrexdmb  5232  ffnfv  5248  fconstfvm  5304  elunirn  5330  isoini  5382  reldm  5735