Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem7 Structured version   Unicode version

Theorem tfrlem7 6673
 Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem7 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem7
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3
21tfrlem6 6672 . 2 recs
31recsfval 6671 . . . . . . . . 9 recs
43eleq2i 2506 . . . . . . . 8 recs
5 eluni 4042 . . . . . . . 8
64, 5bitri 242 . . . . . . 7 recs
73eleq2i 2506 . . . . . . . 8 recs
8 eluni 4042 . . . . . . . 8
97, 8bitri 242 . . . . . . 7 recs
106, 9anbi12i 680 . . . . . 6 recs recs
11 eeanv 1940 . . . . . 6
1210, 11bitr4i 245 . . . . 5 recs recs
13 an4 799 . . . . . . . 8
14 ancom 439 . . . . . . . 8
1513, 14bitri 242 . . . . . . 7
161tfrlem5 6670 . . . . . . . 8
1716imp 420 . . . . . . 7
1815, 17sylbi 189 . . . . . 6
1918exlimivv 1646 . . . . 5
2012, 19sylbi 189 . . . 4 recs recs
2120ax-gen 1556 . . 3 recs recs
2221gen2 1557 . 2 recs recs
23 dffun4 5495 . 2 recs recs recs recs
242, 22, 23mpbir2an 888 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wal 1550  wex 1551   wceq 1653   wcel 1727  cab 2428  wral 2711  wrex 2712  cop 3841  cuni 4039  con0 4610   cres 4909   wrel 4912   wfun 5477   wfn 5478  cfv 5483  recscrecs 6661 This theorem is referenced by:  tfrlem9  6675  tfrlem9a  6676  tfrlem10  6677  tfrlem14  6681  tfrlem16  6683  tfr1a  6684  tfr1  6687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491  df-recs 6662
 Copyright terms: Public domain W3C validator