Theorem fvelrnb 5401

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 2381	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵))
2		19.41v 1841	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3		simpl 108	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → 𝑥 ∈ 𝐴)
4	3	anim1i 336	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝐹 Fn 𝐴))
5	4	ancomd 265	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴))
6		funfvex 5370	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
7	6	funfni 5159	. . . . . . . 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 109	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → (𝐹‘𝑥) = 𝐵)
10	9	eleq1d 2168	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
11	10	adantr 272	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹‘𝑥) ∈ V ↔ 𝐵 ∈ V))
12	8, 11	mpbid 146	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
13	12	exlimiv 1545	. . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
14	2, 13	sylbir 134	. . . 4 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
15	1, 14	sylanb 280	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
16	15	expcom 115	. 2 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V))
17		fnrnfv 5400	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
18	17	eleq2d 2169	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}))
19		eqeq1 2106	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ 𝐵 = (𝐹‘𝑥)))
20		eqcom 2102	. . . . . 6 ⊢ (𝐵 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵)
21	19, 20	syl6bb 195	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝐵))
22	21	rexbidv 2397	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
23	22	elab3g 2788	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
24	18, 23	sylan9bbr 454	. 2 ⊢ (((∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → 𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
25	16, 24	mpancom 416	1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1299  ∃wex 1436   ∈ wcel 1448  {cab 2086  ∃wrex 2376  Vcvv 2641  ran crn 4478   Fn wfn 5054  ‘cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067

This theorem is referenced by:  chfnrn  5463  rexrn  5489  ralrn  5490  elrnrexdmb  5492  ffnfv  5510  fconstfvm  5570  elunirn  5599  isoini  5651  reldm  6014  ordiso2  6835  eldju  6868  ctssdc  6912  uzn0  9191  frec2uzrand  10019  frecuzrdgtcl  10026  frecuzrdgfunlem  10033  uzin2  10599