Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem12 Structured version   Unicode version

Theorem tfrlem12 6642
 Description: Lemma for transfinite recursion. Show is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1
tfrlem.3 recs recs recs
Assertion
Ref Expression
tfrlem12 recs
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem12
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . 6
21tfrlem8 6637 . . . . 5 recs
32a1i 11 . . . 4 recs recs
4 dmexg 5122 . . . 4 recs recs
5 elon2 4584 . . . 4 recs recs recs
63, 4, 5sylanbrc 646 . . 3 recs recs
7 suceloni 4785 . . . 4 recs recs
8 tfrlem.3 . . . . 5 recs recs recs
91, 8tfrlem10 6640 . . . 4 recs recs
101, 8tfrlem11 6641 . . . . . 6 recs recs
1110ralrimiv 2780 . . . . 5 recs recs
12 fveq2 5720 . . . . . . 7
13 reseq2 5133 . . . . . . . 8
1413fveq2d 5724 . . . . . . 7
1512, 14eqeq12d 2449 . . . . . 6
1615cbvralv 2924 . . . . 5 recs recs
1711, 16sylib 189 . . . 4 recs recs
18 fneq2 5527 . . . . . 6 recs recs
19 raleq 2896 . . . . . 6 recs recs
2018, 19anbi12d 692 . . . . 5 recs recs recs
2120rspcev 3044 . . . 4 recs recs recs
227, 9, 17, 21syl12anc 1182 . . 3 recs
236, 22syl 16 . 2 recs
24 snex 4397 . . . . 5 recs recs
25 unexg 4702 . . . . 5 recs recs recs recs recs recs
2624, 25mpan2 653 . . . 4 recs recs recs recs
278, 26syl5eqel 2519 . . 3 recs
28 fneq1 5526 . . . . . 6
29 fveq1 5719 . . . . . . . 8
30 reseq1 5132 . . . . . . . . 9
3130fveq2d 5724 . . . . . . . 8
3229, 31eqeq12d 2449 . . . . . . 7
3332ralbidv 2717 . . . . . 6
3428, 33anbi12d 692 . . . . 5
3534rexbidv 2718 . . . 4
3635, 1elab2g 3076 . . 3
3727, 36syl 16 . 2 recs
3823, 37mpbird 224 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2421  wral 2697  wrex 2698  cvv 2948   cun 3310  csn 3806  cop 3809   word 4572  con0 4573   csuc 4575   cdm 4870   cres 4872   wfn 5441  cfv 5446  recscrecs 6624 This theorem is referenced by:  tfrlem13  6643 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-suc 4579  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