Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem11 Structured version   Visualization version   Unicode version

Theorem tfrlem11 7106
 Description: Lemma for transfinite recursion. Compute the value of . (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1
tfrlem.3 recs recs recs
Assertion
Ref Expression
tfrlem11 recs recs
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem11
StepHypRef Expression
1 elsuci 5489 . 2 recs recs recs
2 tfrlem.1 . . . . . . . . 9
3 tfrlem.3 . . . . . . . . 9 recs recs recs
42, 3tfrlem10 7105 . . . . . . . 8 recs recs
5 fnfun 5673 . . . . . . . 8 recs
64, 5syl 17 . . . . . . 7 recs
7 ssun1 3597 . . . . . . . . 9 recs recs recs recs
87, 3sseqtr4i 3465 . . . . . . . 8 recs
92tfrlem9 7103 . . . . . . . . 9 recs recs recs
10 funssfv 5880 . . . . . . . . . . . 12 recs recs recs
11103expa 1208 . . . . . . . . . . 11 recs recs recs
1211adantrl 722 . . . . . . . . . 10 recs recs recs recs
13 onelss 5465 . . . . . . . . . . . 12 recs recs recs
1413imp 431 . . . . . . . . . . 11 recs recs recs
15 fun2ssres 5623 . . . . . . . . . . . . 13 recs recs recs
16153expa 1208 . . . . . . . . . . . 12 recs recs recs
1716fveq2d 5869 . . . . . . . . . . 11 recs recs recs
1814, 17sylan2 477 . . . . . . . . . 10 recs recs recs recs
1912, 18eqeq12d 2466 . . . . . . . . 9 recs recs recs recs recs
209, 19syl5ibr 225 . . . . . . . 8 recs recs recs recs
218, 20mpanl2 687 . . . . . . 7 recs recs recs
226, 21sylan 474 . . . . . 6 recs recs recs recs
2322exp32 610 . . . . 5 recs recs recs recs
2423pm2.43i 49 . . . 4 recs recs recs
2524pm2.43d 50 . . 3 recs recs
26 opex 4664 . . . . . . . . 9
2726snid 3996 . . . . . . . 8
28 opeq1 4166 . . . . . . . . . . 11 recs recs
2928adantl 468 . . . . . . . . . 10 recs recs recs
30 eqimss 3484 . . . . . . . . . . . . . 14 recs recs
318, 15mp3an2 1352 . . . . . . . . . . . . . 14 recs recs
326, 30, 31syl2an 480 . . . . . . . . . . . . 13 recs recs recs
33 reseq2 5100 . . . . . . . . . . . . . . 15 recs recs recs recs
342tfrlem6 7100 . . . . . . . . . . . . . . . 16 recs
35 resdm 5146 . . . . . . . . . . . . . . . 16 recs recs recs recs
3634, 35ax-mp 5 . . . . . . . . . . . . . . 15 recs recs recs
3733, 36syl6eq 2501 . . . . . . . . . . . . . 14 recs recs recs
3837adantl 468 . . . . . . . . . . . . 13 recs recs recs recs
3932, 38eqtrd 2485 . . . . . . . . . . . 12 recs recs recs
4039fveq2d 5869 . . . . . . . . . . 11 recs recs recs
4140opeq2d 4173 . . . . . . . . . 10 recs recs recs recs recs
4229, 41eqtrd 2485 . . . . . . . . 9 recs recs recs recs
4342sneqd 3980 . . . . . . . 8 recs recs recs recs
4427, 43syl5eleq 2535 . . . . . . 7 recs recs recs recs
45 elun2 3602 . . . . . . 7 recs recs recs recs recs
4644, 45syl 17 . . . . . 6 recs recs recs recs recs
4746, 3syl6eleqr 2540 . . . . 5 recs recs
484adantr 467 . . . . . 6 recs recs recs
49 simpr 463 . . . . . . 7 recs recs recs
50 sucidg 5501 . . . . . . . 8 recs recs recs
5150adantr 467 . . . . . . 7 recs recs recs recs
5249, 51eqeltrd 2529 . . . . . 6 recs recs recs
53 fnopfvb 5906 . . . . . 6 recs recs
5448, 52, 53syl2anc 667 . . . . 5 recs recs
5547, 54mpbird 236 . . . 4 recs recs
5655ex 436 . . 3 recs recs
5725, 56jaod 382 . 2 recs recs recs
581, 57syl5 33 1 recs recs
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wo 370   wa 371   wceq 1444   wcel 1887  cab 2437  wral 2737  wrex 2738   cun 3402   wss 3404  csn 3968  cop 3974   cdm 4834   cres 4836   wrel 4839  con0 5423   csuc 5425   wfun 5576   wfn 5577  cfv 5582  recscrecs 7089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-wrecs 7028  df-recs 7090 This theorem is referenced by:  tfrlem12  7107
 Copyright terms: Public domain W3C validator