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Theorem tfrlem9a 6402
 Description: Lemma for transfinite recursion. Without using ax-rep 4131, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem9a recs recs
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem9a
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . 5
21tfrlem7 6399 . . . 4 recs
3 funfvop 5637 . . . 4 recs recs recs recs
42, 3mpan 651 . . 3 recs recs recs
51recsfval 6397 . . . . 5 recs
65eleq2i 2347 . . . 4 recs recs recs
7 eluni 3830 . . . 4 recs recs
86, 7bitri 240 . . 3 recs recs recs
94, 8sylib 188 . 2 recs recs
10 simprr 733 . . . . . 6 recs recs
111tfrlem3 6393 . . . . . . 7
1211abeq2i 2390 . . . . . 6
1310, 12sylib 188 . . . . 5 recs recs
142a1i 10 . . . . . . . . . 10 recs recs recs
15 simplrr 737 . . . . . . . . . . . 12 recs recs
16 elssuni 3855 . . . . . . . . . . . 12
1715, 16syl 15 . . . . . . . . . . 11 recs recs
1817, 5syl6sseqr 3225 . . . . . . . . . 10 recs recs recs
19 fndm 5343 . . . . . . . . . . . . . 14
2019ad2antll 709 . . . . . . . . . . . . 13 recs recs
21 simprl 732 . . . . . . . . . . . . 13 recs recs
2220, 21eqeltrd 2357 . . . . . . . . . . . 12 recs recs
23 eloni 4402 . . . . . . . . . . . 12
2422, 23syl 15 . . . . . . . . . . 11 recs recs
25 simpll 730 . . . . . . . . . . . 12 recs recs recs
26 fvex 5539 . . . . . . . . . . . . 13 recs
2726a1i 10 . . . . . . . . . . . 12 recs recs recs
28 simplrl 736 . . . . . . . . . . . . 13 recs recs recs
29 df-br 4024 . . . . . . . . . . . . 13 recs recs
3028, 29sylibr 203 . . . . . . . . . . . 12 recs recs recs
31 breldmg 4884 . . . . . . . . . . . 12 recs recs recs
3225, 27, 30, 31syl3anc 1182 . . . . . . . . . . 11 recs recs
33 ordelss 4408 . . . . . . . . . . 11
3424, 32, 33syl2anc 642 . . . . . . . . . 10 recs recs
35 fun2ssres 5295 . . . . . . . . . 10 recs recs recs
3614, 18, 34, 35syl3anc 1182 . . . . . . . . 9 recs recs recs
37 vex 2791 . . . . . . . . . . 11
3837resex 4995 . . . . . . . . . 10
3938a1i 10 . . . . . . . . 9 recs recs
4036, 39eqeltrd 2357 . . . . . . . 8 recs recs recs
4140expr 598 . . . . . . 7 recs recs recs
4241adantrd 454 . . . . . 6 recs recs recs
4342rexlimdva 2667 . . . . 5 recs recs recs
4413, 43mpd 14 . . . 4 recs recs recs
4544ex 423 . . 3 recs recs recs
4645exlimdv 1664 . 2 recs recs recs
479, 46mpd 14 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1528   wceq 1623   wcel 1684  cab 2269  wral 2543  wrex 2544  cvv 2788   wss 3152  cop 3643  cuni 3827   class class class wbr 4023   word 4391  con0 4392   cdm 4689   cres 4691   wfun 5249   wfn 5250  cfv 5255  recscrecs 6387 This theorem is referenced by:  tfrlem15  6408  tfrlem16  6409  rdgseg  6435 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-13 1686  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  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-recs 6388
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