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| Mirrors > Home > MPE Home > Th. List > Mathboxes > limcmptdm | Structured version Visualization version GIF version | ||
| Description: The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| limcmptdm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| limcmptdm.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| limcmptdm.c | ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) | 
| Ref | Expression | 
|---|---|
| limcmptdm | ⊢ (𝜑 → 𝐴 ⊆ ℂ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | limcmptdm.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | limcmptdm.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
| 3 | 1, 2 | dmmptd 6713 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) | 
| 4 | limcmptdm.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) | |
| 5 | limcrcl 25909 | . . . 4 ⊢ (𝐶 ∈ (𝐹 limℂ 𝐷) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ)) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ)) | 
| 7 | 6 | simp2d 1144 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ) | 
| 8 | 3, 7 | eqsstrrd 4019 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 (class class class)co 7431 ℂcc 11153 limℂ climc 25897 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 df-limc 25901 | 
| This theorem is referenced by: neglimc 45662 addlimc 45663 0ellimcdiv 45664 reclimc 45668 | 
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