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Theorem idlvalNEW 25445
 Description: The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
Hypotheses
Ref Expression
idlvalNEW.1
idlvalNEW.2
idlvalNEW.3
idlvalNEW.4
Assertion
Ref Expression
idlvalNEW IdlNEW
Distinct variable groups:   ,,,,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,,)   (,,,)

Proof of Theorem idlvalNEW
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5
2 idlvalNEW.3 . . . . 5
31, 2syl6eqr 2333 . . . 4
43pweqd 3630 . . 3
5 fveq2 5525 . . . . . 6
6 idlvalNEW.4 . . . . . 6
75, 6syl6eqr 2333 . . . . 5
87eleq1d 2349 . . . 4
9 fveq2 5525 . . . . . . . . . . 11
10 idlvalNEW.1 . . . . . . . . . . 11
119, 10syl6eqr 2333 . . . . . . . . . 10
1211oveqd 5875 . . . . . . . . 9
1312eleq1d 2349 . . . . . . . 8
1413ralbidv 2563 . . . . . . 7
1514adantr 451 . . . . . 6
163eleq2d 2350 . . . . . . . . 9
17 fveq2 5525 . . . . . . . . . . . . 13
18 idlvalNEW.2 . . . . . . . . . . . . 13
1917, 18syl6eqr 2333 . . . . . . . . . . . 12
2019oveqd 5875 . . . . . . . . . . 11
2120eleq1d 2349 . . . . . . . . . 10
2219oveqd 5875 . . . . . . . . . . 11
2322eleq1d 2349 . . . . . . . . . 10
2421, 23anbi12d 691 . . . . . . . . 9
2516, 24imbi12d 311 . . . . . . . 8
2625adantr 451 . . . . . . 7
2726ralbidv2 2565 . . . . . 6
2815, 27anbi12d 691 . . . . 5
2928ralbidva 2559 . . . 4
308, 29anbi12d 691 . . 3
314, 30rabeqbidv 2783 . 2
32 df-idlNEW 25444 . 2 IdlNEW
33 fvex 5539 . . . . 5
342, 33eqeltri 2353 . . . 4
3534pwex 4193 . . 3
3635rabex 4165 . 2
3731, 32, 36fvmpt 5602 1 IdlNEW
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  wral 2543  crab 2547  cvv 2788  cpw 3625  cfv 5255  (class class class)co 5858  cbs 13148   cplusg 13208  cmulr 13209  c0g 13400  crg 15337  IdlNEWcidln 25443 This theorem is referenced by:  isidlNEW  25446 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-idlNEW 25444
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