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Theorem isoselem 5402
Description: Lemma for isose 5403. (Contributed by Mario Carneiro, 23-Jun-2015.)
Hypotheses
Ref Expression
isofrlem.1  H  Isom  R ,  S  ,
isofrlem.2  H "  _V
Assertion
Ref Expression
isoselem  R Se  S Se
Distinct variable groups:   ,   ,   , H   ,   , R   , S

Proof of Theorem isoselem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfse2 4641 . . . . . . . . 9  R Se  i^i  `' R " { } 
_V
21biimpi 113 . . . . . . . 8  R Se  i^i  `' R " { } 
_V
32r19.21bi 2401 . . . . . . 7  R Se  i^i  `' R " { }  _V
43expcom 109 . . . . . 6  R Se  i^i  `' R " { }  _V
54adantl 262 . . . . 5  R Se  i^i  `' R " { } 
_V
6 imaeq2 4607 . . . . . . . . . . 11  i^i  `' R " { }  H "  H "  i^i  `' R " { }
76eleq1d 2103 . . . . . . . . . 10  i^i  `' R " { }  H "  _V  H
"  i^i  `' R " { }  _V
87imbi2d 219 . . . . . . . . 9  i^i  `' R " { }  H " 
_V  H "  i^i  `' R " { }  _V
9 isofrlem.2 . . . . . . . . 9  H "  _V
108, 9vtoclg 2607 . . . . . . . 8  i^i  `' R " { } 
_V  H "  i^i  `' R " { }  _V
1110com12 27 . . . . . . 7  i^i  `' R " { } 
_V  H "  i^i  `' R " { }  _V
1211adantr 261 . . . . . 6  i^i  `' R " { }  _V  H "  i^i  `' R " { }  _V
13 isofrlem.1 . . . . . . . 8  H  Isom  R ,  S  ,
14 isoini 5400 . . . . . . . 8  H  Isom  R ,  S  ,  H "  i^i  `' R " { }  i^i  `' S " { H `  }
1513, 14sylan 267 . . . . . . 7  H "  i^i  `' R " { }  i^i  `' S " { H `  }
1615eleq1d 2103 . . . . . 6  H
"  i^i  `' R " { }  _V  i^i  `' S " { H `  }  _V
1712, 16sylibd 138 . . . . 5  i^i  `' R " { }  _V  i^i  `' S " { H `  } 
_V
185, 17syld 40 . . . 4  R Se  i^i  `' S " { H `  } 
_V
1918ralrimdva 2393 . . 3  R Se  i^i  `' S " { H `  } 
_V
20 isof1o 5390 . . . . 5  H 
Isom  R ,  S  ,  H : -1-1-onto->
21 f1ofn 5070 . . . . 5  H : -1-1-onto->  H  Fn
22 sneq 3378 . . . . . . . . 9  H `  { }  { H `  }
2322imaeq2d 4611 . . . . . . . 8  H `  `' S " { }  `' S " { H `  }
2423ineq2d 3132 . . . . . . 7  H `  i^i  `' S " { }  i^i  `' S " { H `  }
2524eleq1d 2103 . . . . . 6  H `  i^i  `' S " { } 
_V  i^i  `' S " { H `  } 
_V
2625ralrn 5248 . . . . 5  H  Fn  ran  H  i^i  `' S " { } 
_V  i^i  `' S " { H `  } 
_V
2713, 20, 21, 264syl 18 . . . 4 
ran  H  i^i  `' S " { }  _V  i^i  `' S " { H `  } 
_V
28 f1ofo 5076 . . . . . 6  H : -1-1-onto->  H : -onto->
29 forn 5052 . . . . . 6  H : -onto->  ran 
H
3013, 20, 28, 294syl 18 . . . . 5  ran  H
3130raleqdv 2505 . . . 4 
ran  H  i^i  `' S " { }  _V  i^i  `' S " { } 
_V
3227, 31bitr3d 179 . . 3  i^i  `' S " { H `  } 
_V  i^i  `' S " { } 
_V
3319, 32sylibd 138 . 2  R Se  i^i  `' S " { } 
_V
34 dfse2 4641 . 2  S Se  i^i  `' S " { } 
_V
3533, 34syl6ibr 151 1  R Se  S Se
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    i^i cin 2910   {csn 3367   Se wse 4055   `'ccnv 4287   ran crn 4289   "cima 4291    Fn wfn 4840   -onto->wfo 4843   -1-1-onto->wf1o 4844   ` cfv 4845    Isom wiso 4846
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-bnd 1396  ax-4 1397  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
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-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-se 4056  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-isom 4854
This theorem is referenced by:  isose  5403
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