Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem9a Structured version   Unicode version

Theorem tfrlem9a 6639
 Description: Lemma for transfinite recursion. Without using ax-rep 4312, 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 6636 . . . 4 recs
3 funfvop 5834 . . . 4 recs recs recs recs
42, 3mpan 652 . . 3 recs recs recs
51recsfval 6634 . . . . 5 recs
65eleq2i 2499 . . . 4 recs recs recs
7 eluni 4010 . . . 4 recs recs
86, 7bitri 241 . . 3 recs recs recs
94, 8sylib 189 . 2 recs recs
10 simprr 734 . . . 4 recs recs
111tfrlem3 6630 . . . . 5
1211abeq2i 2542 . . . 4
1310, 12sylib 189 . . 3 recs recs
142a1i 11 . . . . . . . 8 recs recs recs
15 simplrr 738 . . . . . . . . . 10 recs recs
16 elssuni 4035 . . . . . . . . . 10
1715, 16syl 16 . . . . . . . . 9 recs recs
1817, 5syl6sseqr 3387 . . . . . . . 8 recs recs recs
19 fndm 5536 . . . . . . . . . . . 12
2019ad2antll 710 . . . . . . . . . . 11 recs recs
21 simprl 733 . . . . . . . . . . 11 recs recs
2220, 21eqeltrd 2509 . . . . . . . . . 10 recs recs
23 eloni 4583 . . . . . . . . . 10
2422, 23syl 16 . . . . . . . . 9 recs recs
25 simpll 731 . . . . . . . . . 10 recs recs recs
26 fvex 5734 . . . . . . . . . . 11 recs
2726a1i 11 . . . . . . . . . 10 recs recs recs
28 simplrl 737 . . . . . . . . . . 11 recs recs recs
29 df-br 4205 . . . . . . . . . . 11 recs recs
3028, 29sylibr 204 . . . . . . . . . 10 recs recs recs
31 breldmg 5067 . . . . . . . . . 10 recs recs recs
3225, 27, 30, 31syl3anc 1184 . . . . . . . . 9 recs recs
33 ordelss 4589 . . . . . . . . 9
3424, 32, 33syl2anc 643 . . . . . . . 8 recs recs
35 fun2ssres 5486 . . . . . . . 8 recs recs recs
3614, 18, 34, 35syl3anc 1184 . . . . . . 7 recs recs recs
37 vex 2951 . . . . . . . . 9
3837resex 5178 . . . . . . . 8
3938a1i 11 . . . . . . 7 recs recs
4036, 39eqeltrd 2509 . . . . . 6 recs recs recs
4140expr 599 . . . . 5 recs recs recs
4241adantrd 455 . . . 4 recs recs recs
4342rexlimdva 2822 . . 3 recs recs recs
4413, 43mpd 15 . 2 recs recs recs
459, 44exlimddv 1648 1 recs recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  cab 2421  wral 2697  wrex 2698  cvv 2948   wss 3312  cop 3809  cuni 4007   class class class wbr 4204   word 4572  con0 4573   cdm 4870   cres 4872   wfun 5440   wfn 5441  cfv 5446  recscrecs 6624 This theorem is referenced by:  tfrlem15  6645  tfrlem16  6646  rdgseg  6672 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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-recs 6625
 Copyright terms: Public domain W3C validator