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 4299 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . 4 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6664 | . . 3 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 143 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
5 | eqid 2821 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
6 | eqid 2821 | . . . . 5 ⊢ (mREx‘𝑇) = (mREx‘𝑇) | |
7 | 5, 6, 2 | mrsubff 32759 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑m (mREx‘𝑇))) |
8 | ffun 6517 | . . . 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 32760 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇))) |
11 | 10 | eleq2i 2904 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) |
12 | 11 | biimpi 218 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) |
13 | fvelima 6731 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ ((mREx‘𝑇) ↑m (mVR‘𝑇)))) → ∃𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 586 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8428 | . . . . . . 7 ⊢ (𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) | |
16 | 15 | adantl 484 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶(mREx‘𝑇)) |
17 | ssidd 3990 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → (mVR‘𝑇) ⊆ (mVR‘𝑇)) | |
18 | wrd0 13889 | . . . . . . 7 ⊢ ∅ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) | |
19 | eqid 2821 | . . . . . . . . 9 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
20 | 19, 5, 6 | mrexval 32748 | . . . . . . . 8 ⊢ (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
21 | 20 | adantr 483 | . . . . . . 7 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) |
22 | 18, 21 | eleqtrrid 2920 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ∅ ∈ (mREx‘𝑇)) |
23 | eqid 2821 | . . . . . . 7 ⊢ (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) | |
24 | 19, 5, 6, 2, 23 | mrsubval 32756 | . . . . . 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 1367 | . . . . 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 6113 | . . . . . . 7 ⊢ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅) = ∅ | |
27 | 26 | oveq2i 7167 | . . . . . 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 18025 | . . . . . . 7 ⊢ ∅ = (0g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) |
29 | 28 | gsum0 17894 | . . . . . 6 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ∅) = ∅ |
30 | 27, 29 | eqtri 2844 | . . . . 5 ⊢ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓‘𝑣), 〈“𝑣”〉)) ∘ ∅)) = ∅ |
31 | 25, 30 | syl6eq 2872 | . . . 4 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ((𝑆‘𝑓)‘∅) = ∅) |
32 | fveq1 6669 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘∅) = (𝐹‘∅)) | |
33 | 32 | eqeq1d 2823 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘∅) = ∅ ↔ (𝐹‘∅) = ∅)) |
34 | 31, 33 | syl5ibcom 247 | . . 3 ⊢ ((𝑇 ∈ V ∧ 𝑓 ∈ ((mREx‘𝑇) ↑m (mVR‘𝑇))) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘∅) = ∅)) |
35 | 34 | rexlimdva 3284 | . 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 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ↦ cmpt 5146 ran crn 5556 “ cima 5558 ∘ ccom 5559 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ↑pm cpm 8407 Word cword 13862 〈“cs1 13949 Σg cgsu 16714 freeMndcfrmd 18012 mCNcmcn 32707 mVRcmvar 32708 mRExcmrex 32713 mRSubstcmrsub 32717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-frmd 18014 df-mrex 32733 df-mrsub 32737 |
This theorem is referenced by: mrsubvrs 32769 |
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