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Theorem 2ndconst 5785
Description: The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
2ndconst  V  2nd  |`  { }  X.  : { }  X.  -1-1-onto->

Proof of Theorem 2ndconst
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3477 . . 3  V  { }
2 fo2ndresm 5731 . . 3  { }  2nd  |`  { }  X.  : { }  X.  -onto->
31, 2syl 14 . 2  V  2nd  |`  { }  X.  : { }  X.  -onto->
4 moeq 2710 . . . . . 6  <. ,  >.
54moani 1967 . . . . 5  <. ,  >.
6 vex 2554 . . . . . . . 8 
_V
76brres 4561 . . . . . . 7  2nd  |`  { }  X.  2nd  { }  X.
8 fo2nd 5727 . . . . . . . . . . 11  2nd : _V -onto-> _V
9 fofn 5051 . . . . . . . . . . 11  2nd
: _V -onto-> _V  2nd 
Fn  _V
108, 9ax-mp 7 . . . . . . . . . 10  2nd  Fn  _V
11 vex 2554 . . . . . . . . . 10 
_V
12 fnbrfvb 5157 . . . . . . . . . 10  2nd  Fn  _V  _V  2nd `  2nd
1310, 11, 12mp2an 402 . . . . . . . . 9  2nd `  2nd
1413anbi1i 431 . . . . . . . 8  2nd `  { }  X.  2nd  { }  X.
15 elxp7 5739 . . . . . . . . . . 11  { }  X.  _V  X.  _V  1st `  { }  2nd `
16 eleq1 2097 . . . . . . . . . . . . . . 15  2nd `  2nd `
1716biimpa 280 . . . . . . . . . . . . . 14  2nd `  2nd `
1817adantrl 447 . . . . . . . . . . . . 13  2nd `  1st `  { }  2nd `
1918adantrl 447 . . . . . . . . . . . 12  2nd `  _V  X.  _V  1st `  { }  2nd `
20 elsni 3391 . . . . . . . . . . . . . 14  1st `  { }  1st `
21 eqopi 5740 . . . . . . . . . . . . . . . 16  _V 
X.  _V  1st `  2nd `  <. ,  >.
2221ancom2s 500 . . . . . . . . . . . . . . 15  _V 
X.  _V  2nd `  1st `  <. ,  >.
2322an12s 499 . . . . . . . . . . . . . 14  2nd `  _V  X.  _V  1st `  <. ,  >.
2420, 23sylanr2 385 . . . . . . . . . . . . 13  2nd `  _V  X.  _V  1st `  { }  <. , 
>.
2524adantrrr 456 . . . . . . . . . . . 12  2nd `  _V  X.  _V  1st `  { }  2nd `  <. , 
>.
2619, 25jca 290 . . . . . . . . . . 11  2nd `  _V  X.  _V  1st `  { }  2nd `  <. ,  >.
2715, 26sylan2b 271 . . . . . . . . . 10  2nd `  { }  X.  <. ,  >.
2827adantl 262 . . . . . . . . 9  V  2nd `  { }  X.  <. ,  >.
29 fveq2 5121 . . . . . . . . . . . 12  <. , 
>.  2nd `  2nd ` 
<. ,  >.
30 op2ndg 5720 . . . . . . . . . . . . 13  V  _V  2nd `  <. ,  >.
316, 30mpan2 401 . . . . . . . . . . . 12  V  2nd `  <. , 
>.
3229, 31sylan9eqr 2091 . . . . . . . . . . 11  V  <. ,  >.  2nd `
3332adantrl 447 . . . . . . . . . 10  V  <. ,  >.  2nd `
34 simprr 484 . . . . . . . . . . 11  V  <. ,  >.  <. ,  >.
35 snidg 3392 . . . . . . . . . . . . 13  V  { }
3635adantr 261 . . . . . . . . . . . 12  V  <. ,  >.  { }
37 simprl 483 . . . . . . . . . . . 12  V  <. ,  >.
38 opelxpi 4319 . . . . . . . . . . . 12  { }  <. , 
>.  { }  X.
3936, 37, 38syl2anc 391 . . . . . . . . . . 11  V  <. ,  >.  <. , 
>.  { }  X.
4034, 39eqeltrd 2111 . . . . . . . . . 10  V  <. ,  >.  { }  X.
4133, 40jca 290 . . . . . . . . 9  V  <. ,  >.  2nd `  { }  X.
4228, 41impbida 528 . . . . . . . 8  V  2nd `  { }  X. 
<. ,  >.
4314, 42syl5bbr 183 . . . . . . 7  V  2nd  { }  X. 
<. ,  >.
447, 43syl5bb 181 . . . . . 6  V  2nd  |`  { }  X.  <. , 
>.
4544mobidv 1933 . . . . 5  V  2nd  |`  { }  X.  <. ,  >.
465, 45mpbiri 157 . . . 4  V  2nd  |`  { }  X.
4746alrimiv 1751 . . 3  V  2nd  |`  { }  X.
48 funcnv2 4902 . . 3  Fun  `' 2nd  |`  { }  X.  2nd  |`  { }  X.
4947, 48sylibr 137 . 2  V  Fun  `' 2nd  |`  { }  X.
50 dff1o3 5075 . 2  2nd  |`  { }  X.  : { }  X. 
-1-1-onto->  2nd  |`  { }  X.  : { }  X.  -onto->  Fun  `' 2nd  |`  { }  X.
513, 49, 50sylanbrc 394 1  V  2nd  |`  { }  X.  : { }  X.  -1-1-onto->
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390  wmo 1898   _Vcvv 2551   {csn 3367   <.cop 3370   class class class wbr 3755    X. cxp 4286   `'ccnv 4287    |` cres 4290   Fun wfun 4839    Fn wfn 4840   -onto->wfo 4843   -1-1-onto->wf1o 4844   ` cfv 4845   1stc1st 5707   2ndc2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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