Mathbox for Mario Carneiro |
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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 4234 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | 2 | rnfvprc 6689 | . . . . 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 33141 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
8 | ffun 6526 | . . . 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 33142 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
11 | 10 | eleq2i 2822 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
12 | 11 | biimpi 219 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) |
13 | fvelima 6756 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑m 𝑉))) → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) | |
14 | 9, 12, 13 | syl2anc 587 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹) |
15 | elmapi 8508 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑓:𝑉⟶𝑅) |
17 | ssidd 3910 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ⊆ 𝑉) | |
18 | eldifi 4027 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶) | |
19 | elun1 4076 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉)) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
21 | 20 | adantr 484 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
22 | mrsubcn.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
23 | 22, 5, 6, 2 | mrsubcv 33139 | . . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
24 | 16, 17, 21, 23 | syl3anc 1373 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
25 | eldifn 4028 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 484 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ¬ 𝑋 ∈ 𝑉) |
27 | 26 | iffalsed 4436 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉) = 〈“𝑋”〉) |
28 | 24, 27 | eqtrd 2771 | . . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉) |
29 | fveq1 6694 | . . . . 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 248 | . . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑m 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
32 | 31 | rexlimdva 3193 | . 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑m 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
33 | 14, 32 | mpan9 510 | 1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 ⊆ wss 3853 ∅c0 4223 ifcif 4425 ran crn 5537 “ cima 5539 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 ↑pm cpm 8487 〈“cs1 14117 mCNcmcn 33089 mVRcmvar 33090 mRExcmrex 33095 mRSubstcmrsub 33099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-frmd 18230 df-mrex 33115 df-mrsub 33119 |
This theorem is referenced by: elmrsubrn 33149 mrsubco 33150 mrsubvrs 33151 |
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