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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelrnres | Structured version Visualization version GIF version |
Description: If 𝐴 is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
nelrnres | ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnresss 38360 | . . 3 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 | |
2 | 1 | a1i 11 | . 2 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵) |
3 | id 22 | . 2 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran 𝐵) | |
4 | ssnel 38227 | . 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶)) | |
5 | 2, 3, 4 | syl2anc 691 | 1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ⊆ wss 3540 ran crn 5039 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 |
This theorem is referenced by: sge0sup 39284 sge0less 39285 sge0resplit 39299 |
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