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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 46047 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})) | |
2 | n0i 4334 | . . . . . 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 3147 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅) |
5 | ralnex 3073 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = (◡𝐹 “ {(𝐹‘𝑥)})) | |
6 | eqcom 2740 | . . . . . . 7 ⊢ (∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅) | |
7 | 6 | notbii 320 | . . . . . 6 ⊢ (¬ ∅ = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ¬ (◡𝐹 “ {(𝐹‘𝑥)}) = ∅) |
8 | 7 | ralbii 3094 | . . . . 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 5308 | . . . 4 ⊢ ∅ ∈ V | |
12 | setpreimafvex.p | . . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
13 | 12 | elsetpreimafvb 46052 | . . . 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 3048 | . 2 ⊢ (∅ ∉ 𝑃 ↔ ¬ ∅ ∈ 𝑃) | |
17 | 15, 16 | sylibr 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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