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Unicode version
Theorem svs3 14830
Description: A very concise definition of a subspace of a vector space.
Assertion
Ref Expression
svs3 |- SubVec = {<.x, y>. | (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))})}
Distinct variable group:   x,y,v
Proof of Theorem svs3
StepHypRef Expression
1 df-svs 14828 . 2 |- SubVec = {<.x, y>. | (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))})}
2 resss 4237 . . . . . . . . . . 11 |- ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) C_ (2nd` (2nd` x))
3 sseq1 2637 . . . . . . . . . . 11 |- ((2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) -> ((2nd` (2nd` v)) C_ (2nd` (2nd` x)) <-> ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) C_ (2nd` (2nd` x))))
42, 3mpbiri 211 . . . . . . . . . 10 |- ((2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) -> (2nd` (2nd` v)) C_ (2nd` (2nd` x)))
5 fveq2 4681 . . . . . . . . . . . . . . . . . . . . . . 23 |- ((1st` v) = (1st` x) -> (1st` (1st` v)) = (1st` (1st` x)))
65rneqd 4188 . . . . . . . . . . . . . . . . . . . . . 22 |- ((1st` v) = (1st` x) -> ran (1st` (1st` v)) = ran (1st` (1st` x)))
76eqcoms 1887 . . . . . . . . . . . . . . . . . . . . 21 |- ((1st` x) = (1st` v) -> ran (1st` (1st` v)) = ran (1st` (1st` x)))
8 xpeq1 4016 . . . . . . . . . . . . . . . . . . . . 21 |- (ran (1st` (1st` v)) = ran (1st` (1st` x)) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))
97, 8syl 12 . . . . . . . . . . . . . . . . . . . 20 |- ((1st` x) = (1st` v) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))
109eqeq1d 1892 . . . . . . . . . . . . . . . . . . 19 |- ((1st` x) = (1st` v) -> ((ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)) <-> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v))))
1110imbi2d 674 . . . . . . . . . . . . . . . . . 18 |- ((1st` x) = (1st` v) -> (((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v))) <-> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))))
12 eqid 1884 . . . . . . . . . . . . . . . . . . . . . 22 |- ran (1st` (1st` v)) = ran (1st` (1st` v))
13 eqid 1884 . . . . . . . . . . . . . . . . . . . . . 22 |- (2nd` (2nd` v)) = (2nd` (2nd` v))
14 eqid 1884 . . . . . . . . . . . . . . . . . . . . . 22 |- ran (1st` (2nd` v)) = ran (1st` (2nd` v))
1512, 13, 14vecax2 14797 . . . . . . . . . . . . . . . . . . . . 21 |- (v e. Vec -> (2nd` (2nd` v)):(ran (1st` (1st` v)) X. ran (1st` (2nd` v)))-->ran (1st` (2nd` v)))
16 fdm 4567 . . . . . . . . . . . . . . . . . . . . . 22 |- ((2nd` (2nd` v)):(ran (1st` (1st` v)) X. ran (1st` (2nd` v)))-->ran (1st` (2nd` v)) -> dom (2nd` (2nd` v)) = (ran (1st` (1st` v)) X. ran (1st` (2nd` v))))
1716eqcomd 1889 . . . . . . . . . . . . . . . . . . . . 21 |- ((2nd` (2nd` v)):(ran (1st` (1st` v)) X. ran (1st` (2nd` v)))-->ran (1st` (2nd` v)) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))
1815, 17syl 12 . . . . . . . . . . . . . . . . . . . 20 |- (v e. Vec -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))
1918a1d 15 . . . . . . . . . . . . . . . . . . 19 |- (v e. Vec -> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v))))
2019adantl 424 . . . . . . . . . . . . . . . . . 18 |- ((x e. Vec /\ v e. Vec) -> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` v)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v))))
2111, 20syl5bi 225 . . . . . . . . . . . . . . . . 17 |- ((1st` x) = (1st` v) -> ((x e. Vec /\ v e. Vec) -> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))))
2221eqcoms 1887 . . . . . . . . . . . . . . . 16 |- ((1st` v) = (1st` x) -> ((x e. Vec /\ v e. Vec) -> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))))
2322impcom 378 . . . . . . . . . . . . . . 15 |- (((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) -> ((1st` (2nd` v)) C_ (1st` (2nd` x)) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v))))
2423imp 377 . . . . . . . . . . . . . 14 |- ((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))
2524adantr 425 . . . . . . . . . . . . 13 |- (((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) -> (ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)))
26 reseq2 4219 . . . . . . . . . . . . 13 |- ((ran (1st` (1st` x)) X. ran (1st` (2nd` v))) = dom (2nd` (2nd` v)) -> ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) = ((2nd` (2nd` x)) |` dom (2nd` (2nd` v))))
2725, 26syl 12 . . . . . . . . . . . 12 |- (((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) -> ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) = ((2nd` (2nd` x)) |` dom (2nd` (2nd` v))))
28 funssres 4460 . . . . . . . . . . . . 13 |- ((Fun (2nd` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) -> ((2nd` (2nd` x)) |` dom (2nd` (2nd` v))) = (2nd` (2nd` v)))
29 eqid 1884 . . . . . . . . . . . . . . . . 17 |- ran (1st` (1st` x)) = ran (1st` (1st` x))
30 eqid 1884 . . . . . . . . . . . . . . . . 17 |- (2nd` (2nd` x)) = (2nd` (2nd` x))
31 eqid 1884 . . . . . . . . . . . . . . . . 17 |- ran (1st` (2nd` x)) = ran (1st` (2nd` x))
3229, 30, 31vecax2 14797 . . . . . . . . . . . . . . . 16 |- (x e. Vec -> (2nd` (2nd` x)):(ran (1st` (1st` x)) X. ran (1st` (2nd` x)))-->ran (1st` (2nd` x)))
33 ffun 4565 . . . . . . . . . . . . . . . 16 |- ((2nd` (2nd` x)):(ran (1st` (1st` x)) X. ran (1st` (2nd` x)))-->ran (1st` (2nd` x)) -> Fun (2nd` (2nd` x)))
3432, 33syl 12 . . . . . . . . . . . . . . 15 |- (x e. Vec -> Fun (2nd` (2nd` x)))
3534adantr 425 . . . . . . . . . . . . . 14 |- ((x e. Vec /\ v e. Vec) -> Fun (2nd` (2nd` x)))
3635ad2antrr 440 . . . . . . . . . . . . 13 |- ((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) -> Fun (2nd` (2nd` x)))
3728, 36sylan 497 . . . . . . . . . . . 12 |- (((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) -> ((2nd` (2nd` x)) |` dom (2nd` (2nd` v))) = (2nd` (2nd` v)))
3827, 37eqtr2d 1926 . . . . . . . . . . 11 |- (((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) -> (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))
3938ex 402 . . . . . . . . . 10 |- ((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) -> ((2nd` (2nd` v)) C_ (2nd` (2nd` x)) -> (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))))
404, 39impbid2 576 . . . . . . . . 9 |- ((((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) /\ (1st` (2nd` v)) C_ (1st` (2nd` x))) -> ((2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))) <-> (2nd` (2nd` v)) C_ (2nd` (2nd` x))))
4140pm5.32da 711 . . . . . . . 8 |- (((x e. Vec /\ v e. Vec) /\ (1st` v) = (1st` x)) -> (((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))) <-> ((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))))
4241pm5.32da 711 . . . . . . 7 |- ((x e. Vec /\ v e. Vec) -> (((1st` v) = (1st` x) /\ ((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))) <-> ((1st` v) = (1st` x) /\ ((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))))))
43 3anass 862 . . . . . . 7 |- (((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))) <-> ((1st` v) = (1st` x) /\ ((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))))
44 3anass 862 . . . . . . 7 |- (((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x))) <-> ((1st` v) = (1st` x) /\ ((1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))))
4542, 43, 443bitr4g 614 . . . . . 6 |- ((x e. Vec /\ v e. Vec) -> (((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v))))) <-> ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))))
4645rabbidva 2286 . . . . 5 |- (x e. Vec -> {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))} = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))})
4746eqeq2d 1895 . . . 4 |- (x e. Vec -> (y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))} <-> y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))}))
4847pm5.32i 707 . . 3 |- ((x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))}) <-> (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))}))
4948opabbii 3402 . 2 |- {<.x, y>. | (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) = ((2nd` (2nd` x)) |` (ran (1st` (1st` x)) X. ran (1st` (2nd` v)))))})} = {<.x, y>. | (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))})}
501, 49eqtri 1908 1 |- SubVec = {<.x, y>. | (x e. Vec /\ y = {v e. Vec | ((1st` v) = (1st` x) /\ (1st` (2nd` v)) C_ (1st` (2nd` x)) /\ (2nd` (2nd` v)) C_ (2nd` (2nd` x)))})}
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {crab 2108   C_ wss 2593  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  Fun wfun 3992  -->wf 3994  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  Veccvec 14792  SubVeccsvec 14827
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-vec 14793  df-svs 14828
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