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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 6257 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | limcmptdm.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷)) | |
5 | limcrcl 24037 | . . . 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 1179 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ℂ) |
8 | 3, 7 | eqsstr3d 3865 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 ↦ cmpt 4952 dom cdm 5342 ⟶wf 6119 (class class class)co 6905 ℂcc 10250 limℂ climc 24025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-pm 8125 df-limc 24029 |
This theorem is referenced by: neglimc 40674 addlimc 40675 0ellimcdiv 40676 reclimc 40680 |
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