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Theorem tfrlem11 6290
 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 4351 . 2 recs recs recs
2 tfrlem.1 . . . . . . . . 9
3 tfrlem.3 . . . . . . . . 9 recs recs recs
42, 3tfrlem10 6289 . . . . . . . 8 recs recs
5 fnfun 5198 . . . . . . . 8 recs
64, 5syl 17 . . . . . . 7 recs
7 ssun1 3248 . . . . . . . . 9 recs recs recs recs
87, 3sseqtr4i 3132 . . . . . . . 8 recs
92tfrlem9 6287 . . . . . . . . 9 recs recs recs
10 funssfv 5395 . . . . . . . . . . . 12 recs recs recs
11103expa 1156 . . . . . . . . . . 11 recs recs recs
1211adantrl 699 . . . . . . . . . 10 recs recs recs recs
13 onelss 4327 . . . . . . . . . . . 12 recs recs recs
1413imp 420 . . . . . . . . . . 11 recs recs recs
15 fun2ssres 5152 . . . . . . . . . . . . 13 recs recs recs
16153expa 1156 . . . . . . . . . . . 12 recs recs recs
1716fveq2d 5381 . . . . . . . . . . 11 recs recs recs
1814, 17sylan2 462 . . . . . . . . . 10 recs recs recs recs
1912, 18eqeq12d 2267 . . . . . . . . 9 recs recs recs recs recs
209, 19syl5ibr 214 . . . . . . . 8 recs recs recs recs
218, 20mpanl2 665 . . . . . . 7 recs recs recs
226, 21sylan 459 . . . . . 6 recs recs recs recs
2322exp32 591 . . . . 5 recs recs recs recs
2423pm2.43i 45 . . . 4 recs recs recs
2524pm2.43d 46 . . 3 recs recs
26 opex 4130 . . . . . . . . 9
2726snid 3571 . . . . . . . 8
28 opeq1 3696 . . . . . . . . . . 11 recs recs
2928adantl 454 . . . . . . . . . 10 recs recs recs
30 eqimss 3151 . . . . . . . . . . . . . 14 recs recs
318, 15mp3an2 1270 . . . . . . . . . . . . . 14 recs recs
326, 30, 31syl2an 465 . . . . . . . . . . . . 13 recs recs recs
33 reseq2 4857 . . . . . . . . . . . . . . 15 recs recs recs recs
342tfrlem6 6284 . . . . . . . . . . . . . . . 16 recs
35 resdm 4900 . . . . . . . . . . . . . . . 16 recs recs recs recs
3634, 35ax-mp 10 . . . . . . . . . . . . . . 15 recs recs recs
3733, 36syl6eq 2301 . . . . . . . . . . . . . 14 recs recs recs
3837adantl 454 . . . . . . . . . . . . 13 recs recs recs recs
3932, 38eqtrd 2285 . . . . . . . . . . . 12 recs recs recs
4039fveq2d 5381 . . . . . . . . . . 11 recs recs recs
4140opeq2d 3703 . . . . . . . . . 10 recs recs recs recs recs
4229, 41eqtrd 2285 . . . . . . . . 9 recs recs recs recs
4342sneqd 3557 . . . . . . . 8 recs recs recs recs
4427, 43syl5eleq 2339 . . . . . . 7 recs recs recs recs
45 elun2 3253 . . . . . . 7 recs recs recs recs recs
4644, 45syl 17 . . . . . 6 recs recs recs recs recs
4746, 3syl6eleqr 2344 . . . . 5 recs recs
484adantr 453 . . . . . 6 recs recs recs
49 simpr 449 . . . . . . 7 recs recs recs
50 sucidg 4363 . . . . . . . 8 recs recs recs
5150adantr 453 . . . . . . 7 recs recs recs recs
5249, 51eqeltrd 2327 . . . . . 6 recs recs recs
53 fnopfvb 5416 . . . . . 6 recs recs
5448, 52, 53syl2anc 645 . . . . 5 recs recs
5547, 54mpbird 225 . . . 4 recs recs
5655ex 425 . . 3 recs recs
5725, 56jaod 371 . 2 recs recs recs
581, 57syl5 30 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wo 359   wa 360   wceq 1619   wcel 1621  cab 2239  wral 2509  wrex 2510   cun 3076   wss 3078  csn 3544  cop 3547  con0 4285   csuc 4287   cdm 4580   cres 4582   wrel 4585   wfun 4586   wfn 4587  cfv 4592  recscrecs 6273 This theorem is referenced by:  tfrlem12  6291 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608  df-recs 6274
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