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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsub0 | Structured version Visualization version GIF version |
Description: The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubccat.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
Ref | Expression |
---|---|
mrsub0 | ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4249 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . 4 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6639 | . . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 143 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
5 | eqid 2798 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
6 | eqid 2798 | . . . . 5 ⊢ (mREx‘𝑇) = (mREx‘𝑇) | |
7 | 5, 6, 2 | mrsubff 32872 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑m (mREx‘𝑇))) |
8 | ffun 6490 | . . . 4 ⊢ (𝑆:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑m (mREx‘𝑇)) → Fun 𝑆) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆) |
10 | 5, 6, 2 | mrsubrn 32873 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇))) |
11 | 10 | eleq2i 2881 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) |
12 | 11 | biimpi 219 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) |
13 | fvelima 6706 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) → ∃𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 587 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8411 | . . . . . . 7 ⊢ (𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) |
17 | ssidd 3938 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → (mVR‘𝑇) ⊆ (mVR‘𝑇)) | |
18 | wrd0 13882 | . . . . . . 7 ⊢ ∅ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) | |
19 | eqid 2798 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
20 | 19, 5, 6 | mrexval 32861 | . . . . . . . 8 ⊢ (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
21 | 20 | adantr 484 | . . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
22 | 18, 21 | eleqtrrid 2897 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ∅ ∈ (mREx‘𝑇)) |
23 | eqid 2798 | . . . . . . 7 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) | |
24 | 19, 5, 6, 2, 23 | mrsubval 32869 | . . . . . 6 ⊢ ((𝑓:(mVR‘𝑇)⟶(mREx‘𝑇) ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ ∅ ∈ (mREx‘𝑇)) → ((𝑆‘𝑓)‘∅) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅))) |
25 | 16, 17, 22, 24 | syl3anc 1368 | . . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ((𝑆‘𝑓)‘∅) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅))) |
26 | co02 6080 | . . . . . . 7 ⊢ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅) = ∅ | |
27 | 26 | oveq2i 7146 | . . . . . 6 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ∅) |
28 | 23 | frmd0 18017 | . . . . . . 7 ⊢ ∅ = (0g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) |
29 | 28 | gsum0 17886 | . . . . . 6 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ∅) = ∅ |
30 | 27, 29 | eqtri 2821 | . . . . 5 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅)) = ∅ |
31 | 25, 30 | eqtrdi 2849 | . . . 4 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ((𝑆‘𝑓)‘∅) = ∅) |
32 | fveq1 6644 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘∅) = (𝐹‘∅)) | |
33 | 32 | eqeq1d 2800 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘∅) = ∅ ↔ (𝐹‘∅) = ∅)) |
34 | 31, 33 | syl5ibcom 248 | . . 3 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘∅) = ∅)) |
35 | 34 | rexlimdva 3243 | . 2 ⊢ (𝑇 ∈ V → (∃𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹 → (𝐹‘∅) = ∅)) |
36 | 4, 14, 35 | sylc 65 | 1 ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 ifcif 4425 ↦ cmpt 5110 ran crn 5520 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ↑pm cpm 8390 Word cword 13857 〈“cs1 13940 Σg cgsu 16706 freeMndcfrmd 18004 mCNcmcn 32820 mVRcmvar 32821 mRExcmrex 32826 mRSubstcmrsub 32830 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-frmd 18006 df-mrex 32846 df-mrsub 32850 |
This theorem is referenced by: mrsubvrs 32882 |
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