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Theorem 0nelsetpreimafv 47382
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 47371 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
2 n0i 4339 . . . . . 6 (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
31, 2syl 17 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
43ralrimiva 3145 . . . 4 (𝐹 Fn 𝐴 → ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
5 ralnex 3071 . . . . 5 (∀𝑥𝐴 ¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
6 eqcom 2743 . . . . . . 7 (∅ = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑥)}) = ∅)
76notbii 320 . . . . . 6 (¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
87ralbii 3092 . . . . 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 5306 . . . 4 ∅ ∈ V
12 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1312elsetpreimafvb 47376 . . . 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 3046 . 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 1539  wcel 2107  {cab 2713  wnel 3045  wral 3060  wrex 3069  Vcvv 3479  c0 4332  {csn 4625  ccnv 5683  cima 5687   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  uniimaelsetpreimafv  47388  imasetpreimafvbijlemfv1  47395
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