Theorem 0nelsetpreimafv 47371

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 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

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

Step	Hyp	Ref	Expression
1		preimafvsnel 47360	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2		n0i 4320	. . . . . 6 ⊢ (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
3	1, 2	syl 17	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
4	3	ralrimiva 3133	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
5		ralnex 3063	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
6		eqcom 2743	. . . . . . 7 ⊢ (∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
7	6	notbii 320	. . . . . 6 ⊢ (¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
8	7	ralbii 3083	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
9	5, 8	bitr3i 277	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
10	4, 9	sylibr 234	. . 3 ⊢ (𝐹 Fn 𝐴 → ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
11		0ex 5282	. . . 4 ⊢ ∅ ∈ V
12		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
13	12	elsetpreimafvb 47365	. . . 4 ⊢ (∅ ∈ V → (∅ ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)})))
14	11, 13	ax-mp 5	. . 3 ⊢ (∅ ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
15	10, 14	sylnibr 329	. 2 ⊢ (𝐹 Fn 𝐴 → ¬ ∅ ∈ 𝑃)
16		df-nel 3038	. 2 ⊢ (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃)
17	15, 16	sylibr 234	1 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714   ∉ wnel 3037  ∀wral 3052  ∃wrex 3061  Vcvv 3464  ∅c0 4313  {csn 4606  ^◡ccnv 5658   “ cima 5662   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  uniimaelsetpreimafv  47377  imasetpreimafvbijlemfv1  47384