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Theorem nfiso 5389
Description: Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
nfiso.1  F/_ H
nfiso.2  F/_ R
nfiso.3  F/_ S
nfiso.4  F/_
nfiso.5  F/_
Assertion
Ref Expression
nfiso  F/  H  Isom  R ,  S  ,

Proof of Theorem nfiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 4854 . 2  H 
Isom  R ,  S  ,  H : -1-1-onto->  R  H `
 S H `
2 nfiso.1 . . . 4  F/_ H
3 nfiso.4 . . . 4  F/_
4 nfiso.5 . . . 4  F/_
52, 3, 4nff1o 5067 . . 3  F/  H : -1-1-onto->
6 nfcv 2175 . . . . . . 7  F/_
7 nfiso.2 . . . . . . 7  F/_ R
8 nfcv 2175 . . . . . . 7  F/_
96, 7, 8nfbr 3799 . . . . . 6  F/  R
102, 6nffv 5128 . . . . . . 7  F/_ H `
11 nfiso.3 . . . . . . 7  F/_ S
122, 8nffv 5128 . . . . . . 7  F/_ H `
1310, 11, 12nfbr 3799 . . . . . 6  F/ H `  S H `
149, 13nfbi 1478 . . . . 5  F/ R  H `
 S H `
153, 14nfralxy 2354 . . . 4  F/  R  H `
 S H `
163, 15nfralxy 2354 . . 3  F/  R  H `
 S H `
175, 16nfan 1454 . 2  F/ H : -1-1-onto->  R  H `  S H `
181, 17nfxfr 1360 1  F/  H  Isom  R ,  S  ,
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   F/wnf 1346   F/_wnfc 2162  wral 2300   class class class wbr 3755   -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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  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-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by: (None)
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