Theorem 0nelsetpreimafv 44730

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 44719	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2		n0i 4264	. . . . . 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 3107	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
5		ralnex 3163	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
6		eqcom 2745	. . . . . . 7 ⊢ (∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
7	6	notbii 319	. . . . . 6 ⊢ (¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
8	7	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
9	5, 8	bitr3i 276	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
10	4, 9	sylibr 233	. . 3 ⊢ (𝐹 Fn 𝐴 → ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
11		0ex 5226	. . . 4 ⊢ ∅ ∈ V
12		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
13	12	elsetpreimafvb 44724	. . . 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 328	. 2 ⊢ (𝐹 Fn 𝐴 → ¬ ∅ ∈ 𝑃)
16		df-nel 3049	. 2 ⊢ (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃)
17	15, 16	sylibr 233	1 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ∉ wnel 3048  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ∅c0 4253  {csn 4558  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  uniimaelsetpreimafv  44736  imasetpreimafvbijlemfv1  44743