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Theorem afvelrnb 41564
 Description: A member of a function's range is a value of the function, analogous to fvelrnb 6282 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 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (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.
StepHypRef Expression
1 fnrnafv 41563 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
21adantr 480 . . 3 ((𝐹 Fn 𝐴𝐵𝑉) → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
32eleq2d 2716 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}))
4 eqeq1 2655 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
5 eqcom 2658 . . . . . 6 (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
64, 5syl6bb 276 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
76rexbidv 3081 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
87elabg 3383 . . 3 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
98adantl 481 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
103, 9bitrd 268 1 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  ran crn 5144   Fn wfn 5921  '''cafv 41515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-dfat 41517  df-afv 41518 This theorem is referenced by: (None)
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