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Theorem tfrlem11 6649
 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 4647 . 2 recs recs recs
2 tfrlem.1 . . . . . . . . 9
3 tfrlem.3 . . . . . . . . 9 recs recs recs
42, 3tfrlem10 6648 . . . . . . . 8 recs recs
5 fnfun 5542 . . . . . . . 8 recs
64, 5syl 16 . . . . . . 7 recs
7 ssun1 3510 . . . . . . . . 9 recs recs recs recs
87, 3sseqtr4i 3381 . . . . . . . 8 recs
92tfrlem9 6646 . . . . . . . . 9 recs recs recs
10 funssfv 5746 . . . . . . . . . . . 12 recs recs recs
11103expa 1153 . . . . . . . . . . 11 recs recs recs
1211adantrl 697 . . . . . . . . . 10 recs recs recs recs
13 onelss 4623 . . . . . . . . . . . 12 recs recs recs
1413imp 419 . . . . . . . . . . 11 recs recs recs
15 fun2ssres 5494 . . . . . . . . . . . . 13 recs recs recs
16153expa 1153 . . . . . . . . . . . 12 recs recs recs
1716fveq2d 5732 . . . . . . . . . . 11 recs recs recs
1814, 17sylan2 461 . . . . . . . . . 10 recs recs recs recs
1912, 18eqeq12d 2450 . . . . . . . . 9 recs recs recs recs recs
209, 19syl5ibr 213 . . . . . . . 8 recs recs recs recs
218, 20mpanl2 663 . . . . . . 7 recs recs recs
226, 21sylan 458 . . . . . 6 recs recs recs recs
2322exp32 589 . . . . 5 recs recs recs recs
2423pm2.43i 45 . . . 4 recs recs recs
2524pm2.43d 46 . . 3 recs recs
26 opex 4427 . . . . . . . . 9
2726snid 3841 . . . . . . . 8
28 opeq1 3984 . . . . . . . . . . 11 recs recs
2928adantl 453 . . . . . . . . . 10 recs recs recs
30 eqimss 3400 . . . . . . . . . . . . . 14 recs recs
318, 15mp3an2 1267 . . . . . . . . . . . . . 14 recs recs
326, 30, 31syl2an 464 . . . . . . . . . . . . 13 recs recs recs
33 reseq2 5141 . . . . . . . . . . . . . . 15 recs recs recs recs
342tfrlem6 6643 . . . . . . . . . . . . . . . 16 recs
35 resdm 5184 . . . . . . . . . . . . . . . 16 recs recs recs recs
3634, 35ax-mp 8 . . . . . . . . . . . . . . 15 recs recs recs
3733, 36syl6eq 2484 . . . . . . . . . . . . . 14 recs recs recs
3837adantl 453 . . . . . . . . . . . . 13 recs recs recs recs
3932, 38eqtrd 2468 . . . . . . . . . . . 12 recs recs recs
4039fveq2d 5732 . . . . . . . . . . 11 recs recs recs
4140opeq2d 3991 . . . . . . . . . 10 recs recs recs recs recs
4229, 41eqtrd 2468 . . . . . . . . 9 recs recs recs recs
4342sneqd 3827 . . . . . . . 8 recs recs recs recs
4427, 43syl5eleq 2522 . . . . . . 7 recs recs recs recs
45 elun2 3515 . . . . . . 7 recs recs recs recs recs
4644, 45syl 16 . . . . . 6 recs recs recs recs recs
4746, 3syl6eleqr 2527 . . . . 5 recs recs
484adantr 452 . . . . . 6 recs recs recs
49 simpr 448 . . . . . . 7 recs recs recs
50 sucidg 4659 . . . . . . . 8 recs recs recs
5150adantr 452 . . . . . . 7 recs recs recs recs
5249, 51eqeltrd 2510 . . . . . 6 recs recs recs
53 fnopfvb 5768 . . . . . 6 recs recs
5448, 52, 53syl2anc 643 . . . . 5 recs recs
5547, 54mpbird 224 . . . 4 recs recs
5655ex 424 . . 3 recs recs
5725, 56jaod 370 . 2 recs recs recs
581, 57syl5 30 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  cab 2422  wral 2705  wrex 2706   cun 3318   wss 3320  csn 3814  cop 3817  con0 4581   csuc 4583   cdm 4878   cres 4880   wrel 4883   wfun 5448   wfn 5449  cfv 5454  recscrecs 6632 This theorem is referenced by:  tfrlem12  6650 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-recs 6633
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