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Mirrors > Home > MPE Home > Th. List > wrdexb | Structured version Visualization version GIF version |
Description: The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.) |
Ref | Expression |
---|---|
wrdexb | ⊢ (𝑆 ∈ V ↔ Word 𝑆 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdexg 13315 | . 2 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V) | |
2 | opex 4932 | . . . . . . . 8 ⊢ 〈0, 𝑠〉 ∈ V | |
3 | 2 | snid 4208 | . . . . . . 7 ⊢ 〈0, 𝑠〉 ∈ {〈0, 𝑠〉} |
4 | snopiswrd 13314 | . . . . . . 7 ⊢ (𝑠 ∈ 𝑆 → {〈0, 𝑠〉} ∈ Word 𝑆) | |
5 | elunii 4441 | . . . . . . 7 ⊢ ((〈0, 𝑠〉 ∈ {〈0, 𝑠〉} ∧ {〈0, 𝑠〉} ∈ Word 𝑆) → 〈0, 𝑠〉 ∈ ∪ Word 𝑆) | |
6 | 3, 4, 5 | sylancr 695 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → 〈0, 𝑠〉 ∈ ∪ Word 𝑆) |
7 | c0ex 10034 | . . . . . . 7 ⊢ 0 ∈ V | |
8 | vex 3203 | . . . . . . 7 ⊢ 𝑠 ∈ V | |
9 | 7, 8 | opeluu 4939 | . . . . . 6 ⊢ (〈0, 𝑠〉 ∈ ∪ Word 𝑆 → (0 ∈ ∪ ∪ ∪ Word 𝑆 ∧ 𝑠 ∈ ∪ ∪ ∪ Word 𝑆)) |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝑠 ∈ 𝑆 → (0 ∈ ∪ ∪ ∪ Word 𝑆 ∧ 𝑠 ∈ ∪ ∪ ∪ Word 𝑆)) |
11 | 10 | simprd 479 | . . . 4 ⊢ (𝑠 ∈ 𝑆 → 𝑠 ∈ ∪ ∪ ∪ Word 𝑆) |
12 | 11 | ssriv 3607 | . . 3 ⊢ 𝑆 ⊆ ∪ ∪ ∪ Word 𝑆 |
13 | uniexg 6955 | . . . 4 ⊢ (Word 𝑆 ∈ V → ∪ Word 𝑆 ∈ V) | |
14 | uniexg 6955 | . . . 4 ⊢ (∪ Word 𝑆 ∈ V → ∪ ∪ Word 𝑆 ∈ V) | |
15 | uniexg 6955 | . . . 4 ⊢ (∪ ∪ Word 𝑆 ∈ V → ∪ ∪ ∪ Word 𝑆 ∈ V) | |
16 | 13, 14, 15 | 3syl 18 | . . 3 ⊢ (Word 𝑆 ∈ V → ∪ ∪ ∪ Word 𝑆 ∈ V) |
17 | ssexg 4804 | . . 3 ⊢ ((𝑆 ⊆ ∪ ∪ ∪ Word 𝑆 ∧ ∪ ∪ ∪ Word 𝑆 ∈ V) → 𝑆 ∈ V) | |
18 | 12, 16, 17 | sylancr 695 | . 2 ⊢ (Word 𝑆 ∈ V → 𝑆 ∈ V) |
19 | 1, 18 | impbii 199 | 1 ⊢ (𝑆 ∈ V ↔ Word 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 〈cop 4183 ∪ cuni 4436 0cc0 9936 Word cword 13291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-word 13299 |
This theorem is referenced by: efgrcl 18128 |
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