Theorem 0nelsetpreimafv 46509

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 46498	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
2		n0i 4325	. . . . . 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 3138	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
5		ralnex 3064	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
6		eqcom 2731	. . . . . . 7 ⊢ (∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
7	6	notbii 320	. . . . . 6 ⊢ (¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅)
8	7	ralbii 3085	. . . . 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 233	. . 3 ⊢ (𝐹 Fn 𝐴 → ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)}))
11		0ex 5297	. . . 4 ⊢ ∅ ∈ V
12		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
13	12	elsetpreimafvb 46503	. . . 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 3039	. 2 ⊢ (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃)
17	15, 16	sylibr 233	1 ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701   ∉ wnel 3038  ∀wral 3053  ∃wrex 3062  Vcvv 3466  ∅c0 4314  {csn 4620  ^◡ccnv 5665   “ cima 5669   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is referenced by:  uniimaelsetpreimafv  46515  imasetpreimafvbijlemfv1  46522