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Theorem tfrlem8 5934
 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 recs
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem8
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . . . . 9
21tfrlem3 5926 . . . . . . . 8
32abeq2i 2148 . . . . . . 7
4 fndm 4998 . . . . . . . . . . 11
54adantr 261 . . . . . . . . . 10
65eleq1d 2106 . . . . . . . . 9
76biimprcd 149 . . . . . . . 8
87rexlimiv 2427 . . . . . . 7
93, 8sylbi 114 . . . . . 6
10 eleq1a 2109 . . . . . 6
119, 10syl 14 . . . . 5
1211rexlimiv 2427 . . . 4
1312abssi 3015 . . 3
14 ssorduni 4213 . . 3
1513, 14ax-mp 7 . 2
161recsfval 5931 . . . . 5 recs
1716dmeqi 4536 . . . 4 recs
18 dmuni 4545 . . . 4
19 vex 2560 . . . . . 6
2019dmex 4598 . . . . 5
2120dfiun2 3691 . . . 4
2217, 18, 213eqtri 2064 . . 3 recs
23 ordeq 4109 . . 3 recs recs
2422, 23ax-mp 7 . 2 recs
2515, 24mpbir 134 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  cab 2026  wral 2306  wrex 2307   wss 2917  cuni 3580  ciun 3657   word 4099  con0 4100   cdm 4345   cres 4347   wfn 4897  cfv 4902  recscrecs 5919 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-tr 3855  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920 This theorem is referenced by:  tfrlemi14d  5947
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