Theorem afvelrnb 47789

Description: A member of a function's range is a value of the function, analogous to fvelrnb 6942 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvelrnb	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem afvelrnb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnafv 47788	. . . 4 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
2	1	adantr 485	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})
3	2	eleq2d 2855	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}))
4		eqeq1 2773	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
5		eqcom 2776	. . . . . 6 ⊢ (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
6	4, 5	bitrdi 290	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
7	6	rexbidv 3195	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
8	7	elabg 3644	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
9	8	adantl 486	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))
10	3, 9	bitrd 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  ran crn 5663   Fn wfn 6532  '''cafv 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-aiota 47711  df-dfat 47745  df-afv 47746

This theorem is referenced by: (None)