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Theorem 1stconst 5784
Description: The mapping of a restriction of the  1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
1stconst  V  1st  |`  X.  { } :  X.  { } -1-1-onto->

Proof of Theorem 1stconst
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3477 . . 3  V  { }
2 fo1stresm 5730 . . 3  { }  1st  |`  X.  { } :  X.  { } -onto->
31, 2syl 14 . 2  V  1st  |`  X.  { } :  X.  { } -onto->
4 moeq 2710 . . . . . 6  <. ,  >.
54moani 1967 . . . . 5  <. ,  >.
6 vex 2554 . . . . . . . 8 
_V
76brres 4561 . . . . . . 7  1st  |`  X.  { }  1st  X.  { }
8 fo1st 5726 . . . . . . . . . . 11  1st : _V -onto-> _V
9 fofn 5051 . . . . . . . . . . 11  1st
: _V -onto-> _V  1st 
Fn  _V
108, 9ax-mp 7 . . . . . . . . . 10  1st  Fn  _V
11 vex 2554 . . . . . . . . . 10 
_V
12 fnbrfvb 5157 . . . . . . . . . 10  1st  Fn  _V  _V  1st `  1st
1310, 11, 12mp2an 402 . . . . . . . . 9  1st `  1st
1413anbi1i 431 . . . . . . . 8  1st `  X.  { }  1st  X.  { }
15 elxp7 5739 . . . . . . . . . . 11  X.  { }  _V  X.  _V  1st `  2nd ` 
{ }
16 eleq1 2097 . . . . . . . . . . . . . . 15  1st `  1st `
1716biimpa 280 . . . . . . . . . . . . . 14  1st `  1st `
1817adantrr 448 . . . . . . . . . . . . 13  1st `  1st `  2nd `  { }
1918adantrl 447 . . . . . . . . . . . 12  1st `  _V  X.  _V  1st `  2nd `  { }
20 elsni 3391 . . . . . . . . . . . . . 14  2nd `  { }  2nd `
21 eqopi 5740 . . . . . . . . . . . . . . 15  _V 
X.  _V  1st `  2nd `  <. ,  >.
2221an12s 499 . . . . . . . . . . . . . 14  1st `  _V  X.  _V  2nd `  <. ,  >.
2320, 22sylanr2 385 . . . . . . . . . . . . 13  1st `  _V  X.  _V  2nd `  { }  <. ,  >.
2423adantrrl 455 . . . . . . . . . . . 12  1st `  _V  X.  _V  1st `  2nd `  { }  <. ,  >.
2519, 24jca 290 . . . . . . . . . . 11  1st `  _V  X.  _V  1st `  2nd `  { }  <. ,  >.
2615, 25sylan2b 271 . . . . . . . . . 10  1st `  X.  { }  <. ,  >.
2726adantl 262 . . . . . . . . 9  V  1st `  X.  { }  <. ,  >.
28 simprr 484 . . . . . . . . . . . 12  V  <. ,  >.  <. ,  >.
2928fveq2d 5125 . . . . . . . . . . 11  V  <. ,  >.  1st `  1st ` 
<. ,  >.
30 simprl 483 . . . . . . . . . . . 12  V  <. ,  >.
31 simpl 102 . . . . . . . . . . . 12  V  <. ,  >.  V
32 op1stg 5719 . . . . . . . . . . . 12  V  1st `  <. ,  >.
3330, 31, 32syl2anc 391 . . . . . . . . . . 11  V  <. ,  >.  1st `  <. ,  >.
3429, 33eqtrd 2069 . . . . . . . . . 10  V  <. ,  >.  1st `
35 snidg 3392 . . . . . . . . . . . . 13  V  { }
3635adantr 261 . . . . . . . . . . . 12  V  <. ,  >.  { }
37 opelxpi 4319 . . . . . . . . . . . 12  { }  <. ,  >.  X.  { }
3830, 36, 37syl2anc 391 . . . . . . . . . . 11  V  <. ,  >.  <. ,  >.  X.  { }
3928, 38eqeltrd 2111 . . . . . . . . . 10  V  <. ,  >.  X.  { }
4034, 39jca 290 . . . . . . . . 9  V  <. ,  >.  1st `  X.  { }
4127, 40impbida 528 . . . . . . . 8  V  1st `  X.  { } 
<. ,  >.
4214, 41syl5bbr 183 . . . . . . 7  V  1st  X.  { } 
<. ,  >.
437, 42syl5bb 181 . . . . . 6  V  1st  |`  X.  { }  <. ,  >.
4443mobidv 1933 . . . . 5  V  1st  |`  X.  { }  <. ,  >.
455, 44mpbiri 157 . . . 4  V  1st  |`  X.  { }
4645alrimiv 1751 . . 3  V  1st  |`  X.  { }
47 funcnv2 4902 . . 3  Fun  `' 1st  |`  X.  { }  1st  |`  X.  { }
4846, 47sylibr 137 . 2  V  Fun  `' 1st  |`  X.  { }
49 dff1o3 5075 . 2  1st  |`  X.  { } :  X.  { }
-1-1-onto->  1st  |`  X.  { } :  X.  { } -onto->  Fun  `' 1st  |`  X.  { }
503, 48, 49sylanbrc 394 1  V  1st  |`  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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