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Mathbox for Glauco Siliprandi |
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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 6708 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | limcmptdm.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) | |
5 | limcrcl 25897 | . . . 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 1140 | . 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ↦ cmpt 5238 dom cdm 5684 ⟶wf 6552 (class class class)co 7426 ℂcc 11158 limℂ climc 25885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pm 8860 df-limc 25889 |
This theorem is referenced by: neglimc 45286 addlimc 45287 0ellimcdiv 45288 reclimc 45292 |
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