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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmressn | Structured version Visualization version GIF version |
Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
Ref | Expression |
---|---|
eldmressn | ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3959 | . . 3 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹)) | |
2 | elsni 4640 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴) |
4 | 1, 3 | sylbi 216 | . 2 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴) |
5 | dmres 5997 | . 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹) | |
6 | 4, 5 | eleq2s 2845 | 1 ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 {csn 4623 dom cdm 5669 ↾ cres 5671 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 df-res 5681 |
This theorem is referenced by: dfdfat2 46408 |
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