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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubf | Structured version Visualization version GIF version |
Description: A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubccat.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
mrsubccat.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mrsubf | ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4051 | . . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅) | |
2 | mrsubccat.s | . . . . . . . 8 ⊢ 𝑆 = (mRSubst‘𝑇) | |
3 | fvprc 6334 | . . . . . . . 8 ⊢ (¬ 𝑇 ∈ V → (mRSubst‘𝑇) = ∅) | |
4 | 2, 3 | syl5eq 2794 | . . . . . . 7 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅) |
5 | 4 | rneqd 5496 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ran ∅) |
6 | rn0 5520 | . . . . . 6 ⊢ ran ∅ = ∅ | |
7 | 5, 6 | syl6eq 2798 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅) |
8 | 1, 7 | nsyl2 142 | . . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V) |
9 | eqid 2748 | . . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
10 | mrsubccat.r | . . . . 5 ⊢ 𝑅 = (mREx‘𝑇) | |
11 | 9, 10, 2 | mrsubff 31687 | . . . 4 ⊢ (𝑇 ∈ V → 𝑆:(𝑅 ↑pm (mVR‘𝑇))⟶(𝑅 ↑𝑚 𝑅)) |
12 | frn 6202 | . . . 4 ⊢ (𝑆:(𝑅 ↑pm (mVR‘𝑇))⟶(𝑅 ↑𝑚 𝑅) → ran 𝑆 ⊆ (𝑅 ↑𝑚 𝑅)) | |
13 | 8, 11, 12 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → ran 𝑆 ⊆ (𝑅 ↑𝑚 𝑅)) |
14 | id 22 | . . 3 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ ran 𝑆) | |
15 | 13, 14 | sseldd 3733 | . 2 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹 ∈ (𝑅 ↑𝑚 𝑅)) |
16 | elmapi 8033 | . 2 ⊢ (𝐹 ∈ (𝑅 ↑𝑚 𝑅) → 𝐹:𝑅⟶𝑅) | |
17 | 15, 16 | syl 17 | 1 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1620 ∈ wcel 2127 Vcvv 3328 ⊆ wss 3703 ∅c0 4046 ran crn 5255 ⟶wf 6033 ‘cfv 6037 (class class class)co 6801 ↑𝑚 cmap 8011 ↑pm cpm 8012 mVRcmvar 31636 mRExcmrex 31641 mRSubstcmrsub 31645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7899 df-map 8013 df-pm 8014 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8926 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-2 11242 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-fzo 12631 df-seq 12967 df-hash 13283 df-word 13456 df-concat 13458 df-s1 13459 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-0g 16275 df-gsum 16276 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-frmd 17558 df-mrex 31661 df-mrsub 31665 |
This theorem is referenced by: elmrsubrn 31695 mrsubco 31696 mrsubvrs 31697 msubco 31706 msubvrs 31735 |
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