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 4301 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6666 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
4 | 1, 3 | nsyl2 143 | . . . 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 32761 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
8 | ffun 6519 | . . . 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 32762 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
11 | 10 | eleq2i 2906 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
12 | 11 | biimpi 218 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
13 | fvelima 6733 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 586 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8430 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅) | |
16 | 15 | adantl 484 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑓:𝑉⟶𝑅) |
17 | ssidd 3992 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ⊆ 𝑉) | |
18 | eldifi 4105 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶) | |
19 | elun1 4154 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉)) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
21 | 20 | adantr 483 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
22 | mrsubcn.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
23 | 22, 5, 6, 2 | mrsubcv 32759 | . . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
24 | 16, 17, 21, 23 | syl3anc 1367 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
25 | eldifn 4106 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 483 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ¬ 𝑋 ∈ 𝑉) |
27 | 26 | iffalsed 4480 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉) = 〈“𝑋”〉) |
28 | 24, 27 | eqtrd 2858 | . . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉) |
29 | fveq1 6671 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘〈“𝑋”〉) = (𝐹‘〈“𝑋”〉)) | |
30 | 29 | eqeq1d 2825 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉 ↔ (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
31 | 28, 30 | syl5ibcom 247 | . . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
32 | 31 | rexlimdva 3286 | . 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
33 | 14, 32 | mpan9 509 | 1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 ifcif 4469 ran crn 5558 “ cima 5560 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ↑pm cpm 8409 〈“cs1 13951 mCNcmcn 32709 mVRcmvar 32710 mRExcmrex 32715 mRSubstcmrsub 32719 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-frmd 18016 df-mrex 32735 df-mrsub 32739 |
This theorem is referenced by: elmrsubrn 32769 mrsubco 32770 mrsubvrs 32771 |
Copyright terms: Public domain | W3C validator |