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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmsubrn | Structured version Visualization version GIF version |
Description: Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
elmsubrn.e | ⊢ 𝐸 = (mEx‘𝑇) |
elmsubrn.o | ⊢ 𝑂 = (mRSubst‘𝑇) |
elmsubrn.s | ⊢ 𝑆 = (mSubst‘𝑇) |
Ref | Expression |
---|---|
elmsubrn | ⊢ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
2 | eqid 2778 | . . . . . 6 ⊢ (mREx‘𝑇) = (mREx‘𝑇) | |
3 | elmsubrn.s | . . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇) | |
4 | elmsubrn.e | . . . . . 6 ⊢ 𝐸 = (mEx‘𝑇) | |
5 | elmsubrn.o | . . . . . 6 ⊢ 𝑂 = (mRSubst‘𝑇) | |
6 | 1, 2, 3, 4, 5 | msubffval 32019 | . . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉))) |
7 | 1, 2, 5 | mrsubff 32008 | . . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑𝑚 (mREx‘𝑇))) |
8 | 7 | ffnd 6292 | . . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇))) |
9 | fnfvelrn 6620 | . . . . . . 7 ⊢ ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂) | |
10 | 8, 9 | sylan 575 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂) |
11 | 7 | feqmptd 6509 | . . . . . 6 ⊢ (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂‘𝑔))) |
12 | eqidd 2779 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) | |
13 | fveq1 6445 | . . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2nd ‘𝑒)) = ((𝑂‘𝑔)‘(2nd ‘𝑒))) | |
14 | 13 | opeq2d 4643 | . . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉 = 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉) |
15 | 14 | mpteq2dv 4980 | . . . . . 6 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉)) |
16 | 10, 11, 12, 15 | fmptco 6661 | . . . . 5 ⊢ (𝑇 ∈ V → ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂) = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉))) |
17 | 6, 16 | eqtr4d 2817 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂)) |
18 | 17 | rneqd 5598 | . . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂)) |
19 | rnco 5895 | . . . 4 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂) = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ↾ ran 𝑂) | |
20 | ssid 3842 | . . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂 | |
21 | resmpt 5699 | . . . . . 6 ⊢ (ran 𝑂 ⊆ ran 𝑂 → ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) | |
22 | 20, 21 | ax-mp 5 | . . . . 5 ⊢ ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ↾ ran 𝑂) = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
23 | 22 | rneqi 5597 | . . . 4 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ↾ ran 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
24 | 19, 23 | eqtri 2802 | . . 3 ⊢ ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂) = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
25 | 18, 24 | syl6eq 2830 | . 2 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
26 | mpt0 6267 | . . . . 5 ⊢ (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) = ∅ | |
27 | 26 | eqcomi 2787 | . . . 4 ⊢ ∅ = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
28 | fvprc 6439 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅) | |
29 | 3, 28 | syl5eq 2826 | . . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅) |
30 | 5 | rnfvprc 6440 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅) |
31 | 30 | mpteq1d 4973 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
32 | 27, 29, 31 | 3eqtr4a 2840 | . . 3 ⊢ (¬ 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
33 | 32 | rneqd 5598 | . 2 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
34 | 25, 33 | pm2.61i 177 | 1 ⊢ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 ∅c0 4141 〈cop 4404 ↦ cmpt 4965 ran crn 5356 ↾ cres 5357 ∘ ccom 5359 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 1st c1st 7443 2nd c2nd 7444 ↑𝑚 cmap 8140 ↑pm cpm 8141 mVRcmvar 31957 mRExcmrex 31962 mExcmex 31963 mRSubstcmrsub 31966 mSubstcmsub 31967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-gsum 16489 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-frmd 17773 df-mrex 31982 df-mrsub 31986 df-msub 31987 |
This theorem is referenced by: msubco 32027 msubvrs 32056 |
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