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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 6465 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | limcmptdm.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) | |
5 | limcrcl 24477 | . . . 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 3954 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ↦ cmpt 5110 dom cdm 5519 ⟶wf 6320 (class class class)co 7135 ℂcc 10524 limℂ climc 24465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-limc 24469 |
This theorem is referenced by: neglimc 42289 addlimc 42290 0ellimcdiv 42291 reclimc 42295 |
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