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Theorem 0nelsetpreimafv 47315
Description: The empty set is not an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 6-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
0nelsetpreimafv (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)

Proof of Theorem 0nelsetpreimafv
StepHypRef Expression
1 preimafvsnel 47304 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
2 n0i 4346 . . . . . 6 (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
31, 2syl 17 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
43ralrimiva 3144 . . . 4 (𝐹 Fn 𝐴 → ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
5 ralnex 3070 . . . . 5 (∀𝑥𝐴 ¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
6 eqcom 2742 . . . . . . 7 (∅ = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑥)}) = ∅)
76notbii 320 . . . . . 6 (¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
87ralbii 3091 . . . . 5 (∀𝑥𝐴 ¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
95, 8bitr3i 277 . . . 4 (¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
104, 9sylibr 234 . . 3 (𝐹 Fn 𝐴 → ¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
11 0ex 5313 . . . 4 ∅ ∈ V
12 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1312elsetpreimafvb 47309 . . . 4 (∅ ∈ V → (∅ ∈ 𝑃 ↔ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)})))
1411, 13ax-mp 5 . . 3 (∅ ∈ 𝑃 ↔ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
1510, 14sylnibr 329 . 2 (𝐹 Fn 𝐴 → ¬ ∅ ∈ 𝑃)
16 df-nel 3045 . 2 (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃)
1715, 16sylibr 234 1 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wnel 3044  wral 3059  wrex 3068  Vcvv 3478  c0 4339  {csn 4631  ccnv 5688  cima 5692   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  uniimaelsetpreimafv  47321  imasetpreimafvbijlemfv1  47328
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