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Theorem isocnv2 5395
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
isocnv2  H 
Isom  R ,  S  ,  H  Isom  `' R ,  `' S ,

Proof of Theorem isocnv2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5390 . . 3  H 
Isom  R ,  S  ,  H : -1-1-onto->
2 f1ofn 5070 . . 3  H : -1-1-onto->  H  Fn
31, 2syl 14 . 2  H 
Isom  R ,  S  ,  H  Fn
4 isof1o 5390 . . 3  H 
Isom  `' R ,  `' S ,  H : -1-1-onto->
54, 2syl 14 . 2  H 
Isom  `' R ,  `' S ,  H  Fn
6 vex 2554 . . . . . . . . . 10 
_V
7 vex 2554 . . . . . . . . . 10 
_V
86, 7brcnv 4461 . . . . . . . . 9  `' R  R
98a1i 9 . . . . . . . 8  H  Fn  `' R  R
10 funfvex 5135 . . . . . . . . . . 11  Fun  H  dom  H  H `  _V
1110funfni 4942 . . . . . . . . . 10  H  Fn  H `  _V
1211adantr 261 . . . . . . . . 9  H  Fn  H `  _V
13 funfvex 5135 . . . . . . . . . . 11  Fun  H  dom  H  H `  _V
1413funfni 4942 . . . . . . . . . 10  H  Fn  H `  _V
1514adantlr 446 . . . . . . . . 9  H  Fn  H `  _V
16 brcnvg 4459 . . . . . . . . 9  H `  _V  H `  _V  H `  `' S H `  H `  S H `
1712, 15, 16syl2anc 391 . . . . . . . 8  H  Fn  H `
 `' S H `  H `  S H `
189, 17bibi12d 224 . . . . . . 7  H  Fn  `' R  H `
 `' S H `  R  H `
 S H `
1918ralbidva 2316 . . . . . 6  H  Fn  `' R  H `  `' S H `  R  H `  S H `
2019ralbidva 2316 . . . . 5  H  Fn  `' R  H `  `' S H `  R  H `  S H `
21 ralcom 2467 . . . . 5  R  H `  S H `
 R  H `
 S H `
2220, 21syl6rbbr 188 . . . 4  H  Fn  R  H `
 S H `  `' R  H `  `' S H `
2322anbi2d 437 . . 3  H  Fn  H : -1-1-onto->  R  H `  S H `
 H :
-1-1-onto->  `' R  H `  `' S H `
24 df-isom 4854 . . 3  H 
Isom  R ,  S  ,  H : -1-1-onto->  R  H `
 S H `
25 df-isom 4854 . . 3  H 
Isom  `' R ,  `' S ,  H : -1-1-onto->  `' R  H `  `' S H `
2623, 24, 253bitr4g 212 . 2  H  Fn  H  Isom  R ,  S  ,  H  Isom  `' R ,  `' S ,
273, 5, 26pm5.21nii 619 1  H 
Isom  R ,  S  ,  H  Isom  `' R ,  `' S ,
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wcel 1390  wral 2300   _Vcvv 2551   class class class wbr 3755   `'ccnv 4287    Fn wfn 4840   -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-bndl 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-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-id 4021  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by: (None)
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