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Theorem afvelrnb0 44656
Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6830. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
Assertion
Ref Expression
afvelrnb0 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem afvelrnb0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 44654 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
21eleq2d 2824 . 2 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}))
3 eqeq1 2742 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
4 eqcom 2745 . . . . . 6 (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
53, 4bitrdi 287 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
65rexbidv 3226 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
76elabg 3607 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
87ibi 266 . 2 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵)
92, 8syl6bi 252 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  {cab 2715  wrex 3065  ran crn 5590   Fn wfn 6428  '''cafv 44609
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612
This theorem is referenced by:  ffnafv  44663
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