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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 41445 | . 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 | |
2 | ssnel 41308 | . 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2113 ⊆ wss 3939 ran crn 5559 ↾ cres 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 |
This theorem is referenced by: sge0sup 42680 sge0less 42681 sge0resplit 42695 |
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