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Mirrors > Home > MPE Home > Th. List > Mathboxes > vhmcls | Structured version Visualization version GIF version |
Description: All variable hypotheses are in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mclsval.d | β’ π· = (mDVβπ) |
mclsval.e | β’ πΈ = (mExβπ) |
mclsval.c | β’ πΆ = (mClsβπ) |
mclsval.1 | β’ (π β π β mFS) |
mclsval.2 | β’ (π β πΎ β π·) |
mclsval.3 | β’ (π β π΅ β πΈ) |
ssmclslem.h | β’ π» = (mVHβπ) |
vhmcls.v | β’ π = (mVRβπ) |
vhmcls.3 | β’ (π β π β π) |
Ref | Expression |
---|---|
vhmcls | β’ (π β (π»βπ) β (πΎπΆπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mclsval.d | . . . 4 β’ π· = (mDVβπ) | |
2 | mclsval.e | . . . 4 β’ πΈ = (mExβπ) | |
3 | mclsval.c | . . . 4 β’ πΆ = (mClsβπ) | |
4 | mclsval.1 | . . . 4 β’ (π β π β mFS) | |
5 | mclsval.2 | . . . 4 β’ (π β πΎ β π·) | |
6 | mclsval.3 | . . . 4 β’ (π β π΅ β πΈ) | |
7 | ssmclslem.h | . . . 4 β’ π» = (mVHβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ssmclslem 35084 | . . 3 β’ (π β (π΅ βͺ ran π») β (πΎπΆπ΅)) |
9 | 8 | unssbd 4183 | . 2 β’ (π β ran π» β (πΎπΆπ΅)) |
10 | vhmcls.v | . . . . 5 β’ π = (mVRβπ) | |
11 | 10, 2, 7 | mvhf 35077 | . . . 4 β’ (π β mFS β π»:πβΆπΈ) |
12 | ffn 6710 | . . . 4 β’ (π»:πβΆπΈ β π» Fn π) | |
13 | 4, 11, 12 | 3syl 18 | . . 3 β’ (π β π» Fn π) |
14 | vhmcls.3 | . . 3 β’ (π β π β π) | |
15 | fnfvelrn 7075 | . . 3 β’ ((π» Fn π β§ π β π) β (π»βπ) β ran π») | |
16 | 13, 14, 15 | syl2anc 583 | . 2 β’ (π β (π»βπ) β ran π») |
17 | 9, 16 | sseldd 3978 | 1 β’ (π β (π»βπ) β (πΎπΆπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 ran crn 5670 Fn wfn 6531 βΆwf 6532 βcfv 6536 (class class class)co 7404 mVRcmvar 34980 mExcmex 34986 mDVcmdv 34987 mVHcmvh 34991 mFScmfs 34995 mClscmcls 34996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-frmd 18772 df-mrex 35005 df-mex 35006 df-mrsub 35009 df-msub 35010 df-mvh 35011 df-mpst 35012 df-msr 35013 df-msta 35014 df-mfs 35015 df-mcls 35016 |
This theorem is referenced by: (None) |
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