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Theorem isoini 5400
Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
Assertion
Ref Expression
isoini  H  Isom  R ,  S  ,  D  H "  i^i  `' R " { D }  i^i  `' S " { H `  D }

Proof of Theorem isoini
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . 4  i^i  `' S " { H `  D }  `' S " { H `  D }
2 isof1o 5390 . . . . . . . . 9  H 
Isom  R ,  S  ,  H : -1-1-onto->
3 f1ofo 5076 . . . . . . . . 9  H : -1-1-onto->  H : -onto->
4 forn 5052 . . . . . . . . . 10  H : -onto->  ran 
H
54eleq2d 2104 . . . . . . . . 9  H : -onto->  ran  H
62, 3, 53syl 17 . . . . . . . 8  H 
Isom  R ,  S  ,  ran  H
7 f1ofn 5070 . . . . . . . . 9  H : -1-1-onto->  H  Fn
8 fvelrnb 5164 . . . . . . . . 9  H  Fn  ran  H  H `
92, 7, 83syl 17 . . . . . . . 8  H 
Isom  R ,  S  ,  ran  H  H `
106, 9bitr3d 179 . . . . . . 7  H 
Isom  R ,  S  ,  H `
1110adantr 261 . . . . . 6  H  Isom  R ,  S  ,  D  H `
122, 7syl 14 . . . . . . . 8  H 
Isom  R ,  S  ,  H  Fn
1312anim1i 323 . . . . . . 7  H  Isom  R ,  S  ,  D  H  Fn  D
14 funfvex 5135 . . . . . . . 8  Fun  H  D  dom  H  H `  D  _V
1514funfni 4942 . . . . . . 7  H  Fn  D  H `  D  _V
16 vex 2554 . . . . . . . 8 
_V
1716eliniseg 4638 . . . . . . 7  H `  D  _V  `' S " { H `
 D }  S H `
 D
1813, 15, 173syl 17 . . . . . 6  H  Isom  R ,  S  ,  D  `' S " { H `  D }  S H `
 D
1911, 18anbi12d 442 . . . . 5  H  Isom  R ,  S  ,  D  `' S " { H `  D }  H `  S H `
 D
20 elin 3120 . . . . . . . . . . . 12  i^i  `' R " { D }  `' R " { D }
21 vex 2554 . . . . . . . . . . . . . 14 
_V
2221eliniseg 4638 . . . . . . . . . . . . 13  D  `' R " { D }  R D
2322anbi2d 437 . . . . . . . . . . . 12  D  `' R " { D }  R D
2420, 23syl5bb 181 . . . . . . . . . . 11  D  i^i  `' R " { D }  R D
2524anbi1d 438 . . . . . . . . . 10  D  i^i  `' R " { D }  H  R D  H
26 anass 381 . . . . . . . . . 10  R D  H  R D  H
2725, 26syl6bb 185 . . . . . . . . 9  D  i^i  `' R " { D }  H  R D  H
2827adantl 262 . . . . . . . 8  H  Isom  R ,  S  ,  D  i^i  `' R " { D }  H  R D  H
29 isorel 5391 . . . . . . . . . . . . . 14  H  Isom  R ,  S  ,  D  R D  H `
 S H `  D
30 fnbrfvb 5157 . . . . . . . . . . . . . . . . 17  H  Fn  H `  H
3130bicomd 129 . . . . . . . . . . . . . . . 16  H  Fn  H  H `
3212, 31sylan 267 . . . . . . . . . . . . . . 15  H  Isom  R ,  S  ,  H  H `
3332adantrr 448 . . . . . . . . . . . . . 14  H  Isom  R ,  S  ,  D  H  H `
3429, 33anbi12d 442 . . . . . . . . . . . . 13  H  Isom  R ,  S  ,  D  R D  H  H `  S H `
 D  H `
35 ancom 253 . . . . . . . . . . . . . 14  H `  S H `  D  H `  H `  H `  S H `  D
36 breq1 3758 . . . . . . . . . . . . . . 15  H `  H `  S H `  D  S H `  D
3736pm5.32i 427 . . . . . . . . . . . . . 14  H `  H `  S H `  D  H `  S H `  D
3835, 37bitri 173 . . . . . . . . . . . . 13  H `  S H `  D  H `  H `  S H `  D
3934, 38syl6bb 185 . . . . . . . . . . . 12  H  Isom  R ,  S  ,  D  R D  H  H `  S H `
 D
4039exp32 347 . . . . . . . . . . 11  H 
Isom  R ,  S  ,  D  R D  H  H `  S H `  D
4140com23 72 . . . . . . . . . 10  H 
Isom  R ,  S  ,  D  R D  H  H `  S H `  D
4241imp 115 . . . . . . . . 9  H  Isom  R ,  S  ,  D  R D  H  H `  S H `  D
4342pm5.32d 423 . . . . . . . 8  H  Isom  R ,  S  ,  D  R D  H  H `  S H `
 D
4428, 43bitrd 177 . . . . . . 7  H  Isom  R ,  S  ,  D  i^i  `' R " { D }  H  H `  S H `  D
4544rexbidv2 2323 . . . . . 6  H  Isom  R ,  S  ,  D  i^i  `' R " { D } H  H `
 S H `  D
46 r19.41v 2460 . . . . . 6  H `  S H `  D  H `  S H `  D
4745, 46syl6bb 185 . . . . 5  H  Isom  R ,  S  ,  D  i^i  `' R " { D } H  H `  S H `  D
4819, 47bitr4d 180 . . . 4  H  Isom  R ,  S  ,  D  `' S " { H `  D }  i^i  `' R " { D } H
491, 48syl5bb 181 . . 3  H  Isom  R ,  S  ,  D  i^i  `' S " { H `  D }  i^i  `' R " { D } H
5049abbi2dv 2153 . 2  H  Isom  R ,  S  ,  D  i^i  `' S " { H `  D }  {  |  i^i  `' R " { D } H }
51 dfima2 4613 . 2  H
"  i^i  `' R " { D }  {  |  i^i  `' R " { D } H }
5250, 51syl6reqr 2088 1  H  Isom  R ,  S  ,  D  H "  i^i  `' R " { D }  i^i  `' S " { H `  D }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551    i^i cin 2910   {csn 3367   class class class wbr 3755   `'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-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-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:  isoini2  5401  isoselem  5402
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