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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 4171 | . . 3 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹)) | |
2 | elsni 4586 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴) |
4 | 1, 3 | sylbi 219 | . 2 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴) |
5 | dmres 5877 | . 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹) | |
6 | 4, 5 | eleq2s 2933 | 1 ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 {csn 4569 dom cdm 5557 ↾ cres 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 df-res 5569 |
This theorem is referenced by: dfdfat2 43334 |
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