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Theorem 0nelsetpreimafv 46058
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 46047 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
2 n0i 4334 . . . . . 6 (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
31, 2syl 17 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
43ralrimiva 3147 . . . 4 (𝐹 Fn 𝐴 → ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
5 ralnex 3073 . . . . 5 (∀𝑥𝐴 ¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
6 eqcom 2740 . . . . . . 7 (∅ = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑥)}) = ∅)
76notbii 320 . . . . . 6 (¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
87ralbii 3094 . . . . 5 (∀𝑥𝐴 ¬ ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
95, 8bitr3i 277 . . . 4 (¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}) ↔ ∀𝑥𝐴 ¬ (𝐹 “ {(𝐹𝑥)}) = ∅)
104, 9sylibr 233 . . 3 (𝐹 Fn 𝐴 → ¬ ∃𝑥𝐴 ∅ = (𝐹 “ {(𝐹𝑥)}))
11 0ex 5308 . . . 4 ∅ ∈ V
12 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1312elsetpreimafvb 46052 . . . 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 3048 . 2 (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃)
1715, 16sylibr 233 1 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wnel 3047  wral 3062  wrex 3071  Vcvv 3475  c0 4323  {csn 4629  ccnv 5676  cima 5680   Fn wfn 6539  cfv 6544
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  uniimaelsetpreimafv  46064  imasetpreimafvbijlemfv1  46071
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