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Mirrors > Home > MPE Home > Th. List > lmfun | Structured version Visualization version GIF version |
Description: The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.) |
Ref | Expression |
---|---|
lmfun | ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmrel 20844 | . . 3 ⊢ Rel (⇝𝑡‘𝐽) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐽 ∈ Haus → Rel (⇝𝑡‘𝐽)) |
3 | simpl 472 | . . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ (𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧)) → 𝐽 ∈ Haus) | |
4 | simprl 790 | . . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ (𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧)) → 𝑥(⇝𝑡‘𝐽)𝑦) | |
5 | simprr 792 | . . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ (𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧)) → 𝑥(⇝𝑡‘𝐽)𝑧) | |
6 | 3, 4, 5 | lmmo 20994 | . . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧)) → 𝑦 = 𝑧) |
7 | 6 | ex 449 | . . . . 5 ⊢ (𝐽 ∈ Haus → ((𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧) → 𝑦 = 𝑧)) |
8 | 7 | alrimiv 1842 | . . . 4 ⊢ (𝐽 ∈ Haus → ∀𝑧((𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧) → 𝑦 = 𝑧)) |
9 | 8 | alrimiv 1842 | . . 3 ⊢ (𝐽 ∈ Haus → ∀𝑦∀𝑧((𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧) → 𝑦 = 𝑧)) |
10 | 9 | alrimiv 1842 | . 2 ⊢ (𝐽 ∈ Haus → ∀𝑥∀𝑦∀𝑧((𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧) → 𝑦 = 𝑧)) |
11 | dffun2 5814 | . 2 ⊢ (Fun (⇝𝑡‘𝐽) ↔ (Rel (⇝𝑡‘𝐽) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(⇝𝑡‘𝐽)𝑦 ∧ 𝑥(⇝𝑡‘𝐽)𝑧) → 𝑦 = 𝑧))) | |
12 | 2, 10, 11 | sylanbrc 695 | 1 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 class class class wbr 4583 Rel wrel 5043 Fun wfun 5798 ‘cfv 5804 ⇝𝑡clm 20840 Hauscha 20922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-z 11255 df-uz 11564 df-top 20521 df-topon 20523 df-lm 20843 df-haus 20929 |
This theorem is referenced by: hausmapdom 21113 lmcau 22919 minvecolem4a 27117 minvecolem4b 27118 minvecolem4 27120 hlimf 27478 heiborlem9 32788 bfplem1 32791 |
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