ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvf1olem Unicode version

Theorem cnvf1olem 5787
Description: Lemma for cnvf1o 5788. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1olem  Rel  C  U. `' { }  C  `'  U. `' { C }

Proof of Theorem cnvf1olem
StepHypRef Expression
1 simprr 484 . . . 4  Rel  C  U. `' { } 
C  U. `' { }
2 1st2nd 5749 . . . . . . . 8  Rel  <. 1st `  ,  2nd `  >.
32adantrr 448 . . . . . . 7  Rel  C  U. `' { }  <. 1st `  ,  2nd ` 
>.
43sneqd 3380 . . . . . 6  Rel  C  U. `' { }  { }  { <. 1st `  ,  2nd `  >. }
54cnveqd 4454 . . . . 5  Rel  C  U. `' { }  `' { }  `' { <. 1st `  ,  2nd `  >. }
65unieqd 3582 . . . 4  Rel  C  U. `' { }  U. `' { }  U. `' { <. 1st `  ,  2nd `  >. }
7 1stexg 5736 . . . . . 6  1st ` 
_V
8 2ndexg 5737 . . . . . 6  2nd ` 
_V
9 opswapg 4750 . . . . . 6  1st `  _V  2nd ` 
_V  U. `' { <. 1st `  ,  2nd `  >. }  <. 2nd `  ,  1st `  >.
107, 8, 9syl2anc 391 . . . . 5  U. `' { <. 1st `  ,  2nd `  >. }  <. 2nd `  ,  1st `  >.
1110ad2antrl 459 . . . 4  Rel  C  U. `' { }  U. `' { <. 1st `  ,  2nd `  >. }  <. 2nd `  ,  1st `  >.
121, 6, 113eqtrd 2073 . . 3  Rel  C  U. `' { } 
C  <. 2nd `  ,  1st `  >.
13 simprl 483 . . . . 5  Rel  C  U. `' { }
143, 13eqeltrrd 2112 . . . 4  Rel  C  U. `' { }  <. 1st `  ,  2nd `  >.
15 opelcnvg 4458 . . . . . 6  2nd `  _V  1st ` 
_V  <. 2nd `  ,  1st `  >.  `' 
<. 1st `  ,  2nd `  >.
168, 7, 15syl2anc 391 . . . . 5  <. 2nd `  ,  1st `  >.  `' 
<. 1st `  ,  2nd `  >.
1716ad2antrl 459 . . . 4  Rel  C  U. `' { }  <. 2nd `  ,  1st `  >.  `' 
<. 1st `  ,  2nd `  >.
1814, 17mpbird 156 . . 3  Rel  C  U. `' { }  <. 2nd `  ,  1st `  >.  `'
1912, 18eqeltrd 2111 . 2  Rel  C  U. `' { } 
C  `'
20 opswapg 4750 . . . . . 6  2nd `  _V  1st ` 
_V  U. `' { <. 2nd `  ,  1st `  >. }  <. 1st `  ,  2nd `  >.
218, 7, 20syl2anc 391 . . . . 5  U. `' { <. 2nd `  ,  1st `  >. }  <. 1st `  ,  2nd `  >.
2221eqcomd 2042 . . . 4  <. 1st `  ,  2nd `  >.  U. `' { <. 2nd `  ,  1st `  >. }
2322ad2antrl 459 . . 3  Rel  C  U. `' { }  <. 1st `  ,  2nd `  >.  U. `' { <. 2nd `  ,  1st `  >. }
2412sneqd 3380 . . . . 5  Rel  C  U. `' { }  { C }  { <. 2nd `  ,  1st `  >. }
2524cnveqd 4454 . . . 4  Rel  C  U. `' { }  `' { C }  `' { <. 2nd `  ,  1st `  >. }
2625unieqd 3582 . . 3  Rel  C  U. `' { }  U. `' { C }  U. `' { <. 2nd `  ,  1st `  >. }
2723, 3, 263eqtr4d 2079 . 2  Rel  C  U. `' { }  U. `' { C }
2819, 27jca 290 1  Rel  C  U. `' { }  C  `'  U. `' { C }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   _Vcvv 2551   {csn 3367   <.cop 3370   U.cuni 3571   `'ccnv 4287   Rel wrel 4293   ` 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-v 2553  df-sbc 2759  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-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-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  cnvf1o  5788
  Copyright terms: Public domain W3C validator