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Theorem afvelrnb 44088
 Description: A member of a function's range is a value of the function, analogous to fvelrnb 6715 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 44087 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
21adantr 485 . . 3 ((𝐹 Fn 𝐴𝐵𝑉) → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
32eleq2d 2838 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}))
4 eqeq1 2763 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
5 eqcom 2766 . . . . . 6 (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
64, 5syl6bb 291 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
76rexbidv 3222 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
87elabg 3588 . . 3 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
98adantl 486 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
103, 9bitrd 282 1 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  ∃wrex 3072  ran crn 5526   Fn wfn 6331  '''cafv 44042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-aiota 44009  df-dfat 44044  df-afv 44045 This theorem is referenced by: (None)
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