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Theorem wfrlem9 25538
 Description: Lemma for well-founded recursion. If , then its predecessor class is a subset of . (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem6.1 wrecs
Assertion
Ref Expression
wfrlem9

Proof of Theorem wfrlem9
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem6.1 . . . . . . . 8 wrecs
2 df-wrecs 25523 . . . . . . . 8 wrecs
31, 2eqtri 2455 . . . . . . 7
43dmeqi 5063 . . . . . 6
5 dmuni 5071 . . . . . 6
64, 5eqtri 2455 . . . . 5
76eleq2i 2499 . . . 4
8 eliun 4089 . . . 4
97, 8bitri 241 . . 3
10 eqid 2435 . . . . . . . 8
1110wfrlem1 25530 . . . . . . 7
1211abeq2i 2542 . . . . . 6
13 predeq3 25435 . . . . . . . . . . . . 13
1413sseq1d 3367 . . . . . . . . . . . 12
1514rspccv 3041 . . . . . . . . . . 11
1615adantl 453 . . . . . . . . . 10
17 fndm 5536 . . . . . . . . . . . . 13
1817eleq2d 2502 . . . . . . . . . . . 12
1917sseq2d 3368 . . . . . . . . . . . 12
2018, 19imbi12d 312 . . . . . . . . . . 11
2120adantr 452 . . . . . . . . . 10
2216, 21mpbird 224 . . . . . . . . 9
2322adantrl 697 . . . . . . . 8
24233adant3 977 . . . . . . 7
2524exlimiv 1644 . . . . . 6
2612, 25sylbi 188 . . . . 5
2726reximia 2803 . . . 4
28 ssiun 4125 . . . 4
2927, 28syl 16 . . 3
309, 29sylbi 188 . 2
3130, 6syl6sseqr 3387 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1550   wceq 1652   wcel 1725  cab 2421  wral 2697  wrex 2698   wss 3312  cuni 4007  ciun 4085   cdm 4870   cres 4872   wfn 5441  cfv 5446  cpred 25430  wrecscwrecs 25522 This theorem is referenced by:  wfrlem10  25539  wfrlem14  25543  wfrlem15  25544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-pred 25431  df-wrecs 25523
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