![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubcn | Structured version Visualization version GIF version |
Description: A substitution does not change the value of constant substrings. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubccat.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
mrsubccat.r | ⊢ 𝑅 = (mREx‘𝑇) |
mrsubcn.v | ⊢ 𝑉 = (mVR‘𝑇) |
mrsubcn.c | ⊢ 𝐶 = (mCN‘𝑇) |
Ref | Expression |
---|---|
mrsubcn | ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4293 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6836 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 141 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
5 | mrsubcn.v | . . . . 5 ⊢ 𝑉 = (mVR‘𝑇) | |
6 | mrsubccat.r | . . . . 5 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6, 2 | mrsubff 34106 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
8 | ffun 6671 | . . . 4 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅) → Fun 𝑆) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆) |
10 | 5, 6, 2 | mrsubrn 34107 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
11 | 10 | eleq2i 2829 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
12 | 11 | biimpi 215 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
13 | fvelima 6908 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8787 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅) | |
16 | 15 | adantl 482 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑓:𝑉⟶𝑅) |
17 | ssidd 3967 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ⊆ 𝑉) | |
18 | eldifi 4086 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶) | |
19 | elun1 4136 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉)) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
21 | 20 | adantr 481 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
22 | mrsubcn.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
23 | 22, 5, 6, 2 | mrsubcv 34104 | . . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
24 | 16, 17, 21, 23 | syl3anc 1371 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
25 | eldifn 4087 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 481 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ¬ 𝑋 ∈ 𝑉) |
27 | 26 | iffalsed 4497 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉) = 〈“𝑋”〉) |
28 | 24, 27 | eqtrd 2776 | . . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉) |
29 | fveq1 6841 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘〈“𝑋”〉) = (𝐹‘〈“𝑋”〉)) | |
30 | 29 | eqeq1d 2738 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉 ↔ (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
31 | 28, 30 | syl5ibcom 244 | . . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
32 | 31 | rexlimdva 3152 | . 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
33 | 14, 32 | mpan9 507 | 1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 ifcif 4486 ran crn 5634 “ cima 5636 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 ↑pm cpm 8766 〈“cs1 14483 mCNcmcn 34054 mVRcmvar 34055 mRExcmrex 34060 mRSubstcmrsub 34064 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-0g 17323 df-gsum 17324 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-frmd 18659 df-mrex 34080 df-mrsub 34084 |
This theorem is referenced by: elmrsubrn 34114 mrsubco 34115 mrsubvrs 34116 |
Copyright terms: Public domain | W3C validator |