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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0nelsetpreimafv | Structured version Visualization version GIF version | ||
| Description: The empty set is not an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 6-Mar-2024.) | 
| Ref | Expression | 
|---|---|
| setpreimafvex.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | 
| Ref | Expression | 
|---|---|
| 0nelsetpreimafv | ⊢ (𝐹 Fn 𝐴 → ∅ ∉ 𝑃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | preimafvsnel 47371 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})) | |
| 2 | n0i 4339 | . . . . . 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 3145 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅) | 
| 5 | ralnex 3071 | . . . . 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 3092 | . . . . 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 5306 | . . . 4 ⊢ ∅ ∈ V | |
| 12 | setpreimafvex.p | . . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
| 13 | 12 | elsetpreimafvb 47376 | . . . 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 3046 | . 2 ⊢ (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃) | |
| 17 | 15, 16 | sylibr 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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