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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 3879 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . . . 8 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | fvprc 6097 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mRSubst‘𝑇) = ∅) | |
4 | 2, 3 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅) |
5 | 4 | rneqd 5274 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran ∅) |
6 | rn0 5298 | . . . . . 6 ⊢ ran ∅ = ∅ | |
7 | 5, 6 | syl6eq 2660 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
8 | 1, 7 | nsyl2 141 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
9 | mrsubcn.v | . . . . 5 ⊢ 𝑉 = (mVR‘𝑇) | |
10 | mrsubccat.r | . . . . 5 ⊢ 𝑅 = (mREx‘𝑇) | |
11 | 9, 10, 2 | mrsubff 30663 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅)) |
12 | ffun 5961 | . . . 4 ⊢ (𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑𝑚 𝑅) → Fun 𝑆) | |
13 | 8, 11, 12 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → Fun 𝑆) |
14 | 9, 10, 2 | mrsubrn 30664 | . . . . 5 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑𝑚 𝑉)) |
15 | 14 | eleq2i 2680 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))) |
16 | 15 | biimpi 205 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))) |
17 | fvelima 6158 | . . 3 ⊢ ((Fun 𝑆 ∧ 𝐹 ∈ (𝑆 “ (𝑅 ↑𝑚 𝑉))) → ∃𝑓 ∈ (𝑅 ↑𝑚 𝑉)(𝑆‘𝑓) = 𝐹) | |
18 | 13, 16, 17 | syl2anc 691 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅 ↑𝑚 𝑉)(𝑆‘𝑓) = 𝐹) |
19 | elmapi 7765 | . . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑𝑚 𝑉) → 𝑓:𝑉⟶𝑅) | |
20 | 19 | adantl 481 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑓:𝑉⟶𝑅) |
21 | ssid 3587 | . . . . . . 7 ⊢ 𝑉 ⊆ 𝑉 | |
22 | 21 | a1i 11 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑉 ⊆ 𝑉) |
23 | eldifi 3694 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ 𝐶) | |
24 | elun1 3742 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝐶 ∪ 𝑉)) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
26 | 25 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → 𝑋 ∈ (𝐶 ∪ 𝑉)) |
27 | mrsubcn.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
28 | 27, 9, 10, 2 | mrsubcv 30661 | . . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
29 | 20, 22, 26, 28 | syl3anc 1318 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉)) |
30 | eldifn 3695 | . . . . . . 7 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → ¬ 𝑋 ∈ 𝑉) | |
31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → ¬ 𝑋 ∈ 𝑉) |
32 | 31 | iffalsed 4047 | . . . . 5 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → if(𝑋 ∈ 𝑉, (𝑓‘𝑋), 〈“𝑋”〉) = 〈“𝑋”〉) |
33 | 29, 32 | eqtrd 2644 | . . . 4 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉) |
34 | fveq1 6102 | . . . . 5 ⊢ ((𝑆‘𝑓) = 𝐹 → ((𝑆‘𝑓)‘〈“𝑋”〉) = (𝐹‘〈“𝑋”〉)) | |
35 | 34 | eqeq1d 2612 | . . . 4 ⊢ ((𝑆‘𝑓) = 𝐹 → (((𝑆‘𝑓)‘〈“𝑋”〉) = 〈“𝑋”〉 ↔ (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
36 | 33, 35 | syl5ibcom 234 | . . 3 ⊢ ((𝑋 ∈ (𝐶 ∖ 𝑉) ∧ 𝑓 ∈ (𝑅 ↑𝑚 𝑉)) → ((𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
37 | 36 | rexlimdva 3013 | . 2 ⊢ (𝑋 ∈ (𝐶 ∖ 𝑉) → (∃𝑓 ∈ (𝑅 ↑𝑚 𝑉)(𝑆‘𝑓) = 𝐹 → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉)) |
38 | 18, 37 | mpan9 485 | 1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ran crn 5039 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ↑pm cpm 7745 〈“cs1 13149 mCNcmcn 30611 mVRcmvar 30612 mRExcmrex 30617 mRSubstcmrsub 30621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-frmd 17209 df-mrex 30637 df-mrsub 30641 |
This theorem is referenced by: elmrsubrn 30671 mrsubco 30672 mrsubvrs 30673 |
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