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Theorem tfrlem8 7072
 Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
Hypothesis
Ref Expression
tfrlem.1
Assertion
Ref Expression
tfrlem8
Distinct variable group:   ,,,

Proof of Theorem tfrlem8
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . . . . 9
21tfrlem3 7066 . . . . . . . 8
32abeq2i 2584 . . . . . . 7
4 fndm 5685 . . . . . . . . . . 11
54adantr 465 . . . . . . . . . 10
65eleq1d 2526 . . . . . . . . 9
76biimprcd 225 . . . . . . . 8
87rexlimiv 2943 . . . . . . 7
93, 8sylbi 195 . . . . . 6
10 eleq1a 2540 . . . . . 6
119, 10syl 16 . . . . 5
1211rexlimiv 2943 . . . 4
1312abssi 3574 . . 3
14 ssorduni 6621 . . 3
1513, 14ax-mp 5 . 2
161recsfval 7069 . . . . 5
1716dmeqi 5209 . . . 4
18 dmuni 5217 . . . 4
19 vex 3112 . . . . . 6
2019dmex 6733 . . . . 5
2120dfiun2 4364 . . . 4
2217, 18, 213eqtri 2490 . . 3
23 ordeq 4890 . . 3
2422, 23ax-mp 5 . 2
2515, 24mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  C_wss 3475  U.cuni 4249  U_ciun 4330  Ordword 4882   con0 4883  domcdm 5004  |cres 5006  Fnwfn 5588  cfv 5593  recscrecs 7060 This theorem is referenced by:  tfrlem10  7075  tfrlem12  7077  tfrlem13  7078  tfrlem14  7079  tfrlem15  7080  tfrlem16  7081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-recs 7061
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