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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 2734 | . . . . . 6 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
2 | eqid 2734 | . . . . . 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 35507 | . . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉))) |
7 | 1, 2, 5 | mrsubff 35496 | . . . . . . . 8 ⊢ (𝑇 ∈ V → 𝑂:((mREx‘𝑇) ↑pm (mVR‘𝑇))⟶((mREx‘𝑇) ↑m (mREx‘𝑇))) |
8 | 7 | ffnd 6737 | . . . . . . 7 ⊢ (𝑇 ∈ V → 𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇))) |
9 | fnfvelrn 7099 | . . . . . . 7 ⊢ ((𝑂 Fn ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂) | |
10 | 8, 9 | sylan 580 | . . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇))) → (𝑂‘𝑔) ∈ ran 𝑂) |
11 | 7 | feqmptd 6976 | . . . . . 6 ⊢ (𝑇 ∈ V → 𝑂 = (𝑔 ∈ ((mREx‘𝑇) ↑pm (mVR‘𝑇)) ↦ (𝑂‘𝑔))) |
12 | eqidd 2735 | . . . . . 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 6905 | . . . . . . . 8 ⊢ (𝑓 = (𝑂‘𝑔) → (𝑓‘(2nd ‘𝑒)) = ((𝑂‘𝑔)‘(2nd ‘𝑒))) | |
14 | 13 | opeq2d 4884 | . . . . . . 7 ⊢ (𝑓 = (𝑂‘𝑔) → 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉 = 〈(1st ‘𝑒), ((𝑂‘𝑔)‘(2nd ‘𝑒))〉) |
15 | 14 | mpteq2dv 5249 | . . . . . 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 7148 | . . . . 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 2777 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆 = ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂)) |
18 | 17 | rneqd 5951 | . . 3 ⊢ (𝑇 ∈ V → ran 𝑆 = ran ((𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) ∘ 𝑂)) |
19 | rnco 6273 | . . . 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 4017 | . . . . . 6 ⊢ ran 𝑂 ⊆ ran 𝑂 | |
21 | resmpt 6056 | . . . . . 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 5950 | . . . 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 2762 | . . 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 | eqtrdi 2790 | . 2 ⊢ (𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
26 | mpt0 6710 | . . . . 5 ⊢ (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) = ∅ | |
27 | 26 | eqcomi 2743 | . . . 4 ⊢ ∅ = (𝑓 ∈ ∅ ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) |
28 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅) | |
29 | 3, 28 | eqtrid 2786 | . . . 4 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅) |
30 | 5 | rnfvprc 6900 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑂 = ∅) |
31 | 30 | mpteq1d 5242 | . . . 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 2800 | . . 3 ⊢ (¬ 𝑇 ∈ V → 𝑆 = (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
33 | 32 | rneqd 5951 | . 2 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉))) |
34 | 25, 33 | pm2.61i 182 | 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 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 〈cop 4636 ↦ cmpt 5230 ran crn 5689 ↾ cres 5690 ∘ ccom 5692 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 ↑m cmap 8864 ↑pm cpm 8865 mVRcmvar 35445 mRExcmrex 35450 mExcmex 35451 mRSubstcmrsub 35454 mSubstcmsub 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-frmd 18874 df-mrex 35470 df-mrsub 35474 df-msub 35475 |
This theorem is referenced by: msubco 35515 msubvrs 35544 |
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