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Theorem tfrlem8 5967
 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 5960 . . . . . . . 8
32abeq2i 2190 . . . . . . 7
4 fndm 5029 . . . . . . . . . . 11
54adantr 270 . . . . . . . . . 10
65eleq1d 2148 . . . . . . . . 9
76biimprcd 158 . . . . . . . 8
87rexlimiv 2472 . . . . . . 7
93, 8sylbi 119 . . . . . 6
10 eleq1a 2151 . . . . . 6
119, 10syl 14 . . . . 5
1211rexlimiv 2472 . . . 4
1312abssi 3070 . . 3
14 ssorduni 4239 . . 3
1513, 14ax-mp 7 . 2
161recsfval 5964 . . . . 5 recs
1716dmeqi 4564 . . . 4 recs
18 dmuni 4573 . . . 4
19 vex 2605 . . . . . 6
2019dmex 4626 . . . . 5
2120dfiun2 3720 . . . 4
2217, 18, 213eqtri 2106 . . 3 recs
23 ordeq 4135 . . 3 recs recs
2422, 23ax-mp 7 . 2 recs
2515, 24mpbir 144 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wceq 1285   wcel 1434  cab 2068  wral 2349  wrex 2350   wss 2974  cuni 3609  ciun 3686   word 4125  con0 4126   cdm 4371   cres 4373   wfn 4927  cfv 4932  recscrecs 5953 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-tr 3884  df-iord 4129  df-on 4131  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954 This theorem is referenced by:  tfrlemi14d  5982  tfri1dALT  6000
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