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Theorem tfrlem8 6647
 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 6640 . . . . . . . 8
32abeq2i 2545 . . . . . . 7
4 fndm 5546 . . . . . . . . . . 11
54adantr 453 . . . . . . . . . 10
65eleq1d 2504 . . . . . . . . 9
76biimprcd 218 . . . . . . . 8
87rexlimiv 2826 . . . . . . 7
93, 8sylbi 189 . . . . . 6
10 eleq1a 2507 . . . . . 6
119, 10syl 16 . . . . 5
1211rexlimiv 2826 . . . 4
1312abssi 3420 . . 3
14 ssorduni 4768 . . 3
1513, 14ax-mp 8 . 2
161recsfval 6644 . . . . 5 recs
1716dmeqi 5073 . . . 4 recs
18 dmuni 5081 . . . 4
19 vex 2961 . . . . . 6
2019dmex 5134 . . . . 5
2120dfiun2 4127 . . . 4
2217, 18, 213eqtri 2462 . . 3 recs
23 ordeq 4590 . . 3 recs recs
2422, 23ax-mp 8 . 2 recs
2515, 24mpbir 202 1 recs
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  cab 2424  wral 2707  wrex 2708   wss 3322  cuni 4017  ciun 4095   word 4582  con0 4583   cdm 4880   cres 4882   wfn 5451  cfv 5456  recscrecs 6634 This theorem is referenced by:  tfrlem10  6650  tfrlem12  6652  tfrlem13  6653  tfrlem14  6654  tfrlem15  6655  tfrlem16  6656 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703 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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-recs 6635
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