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Theorem wfrlem9 25478
 Description: Lemma for well-founded recursion. If , then its predecessor class is a subset of . (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem6.1
wfrlem6.2
Assertion
Ref Expression
wfrlem9
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem wfrlem9
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem6.2 . . . . . . 7
21dmeqi 5030 . . . . . 6
3 dmuni 5038 . . . . . 6
42, 3eqtri 2424 . . . . 5
54eleq2i 2468 . . . 4
6 eliun 4057 . . . 4
75, 6bitri 241 . . 3
8 wfrlem6.1 . . . . . . . 8
98wfrlem1 25470 . . . . . . 7
109abeq2i 2511 . . . . . 6
11 predeq3 25385 . . . . . . . . . . . . 13
1211sseq1d 3335 . . . . . . . . . . . 12
1312rspccv 3009 . . . . . . . . . . 11
1413adantl 453 . . . . . . . . . 10
15 fndm 5503 . . . . . . . . . . . . 13
1615eleq2d 2471 . . . . . . . . . . . 12
1715sseq2d 3336 . . . . . . . . . . . 12
1816, 17imbi12d 312 . . . . . . . . . . 11
1918adantr 452 . . . . . . . . . 10
2014, 19mpbird 224 . . . . . . . . 9
2120adantrl 697 . . . . . . . 8
22213adant3 977 . . . . . . 7
2322exlimiv 1641 . . . . . 6
2410, 23sylbi 188 . . . . 5
2524reximia 2771 . . . 4
26 ssiun 4093 . . . 4
2725, 26syl 16 . . 3
287, 27sylbi 188 . 2
2928, 4syl6sseqr 3355 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1547   wceq 1649   wcel 1721  cab 2390  wral 2666  wrex 2667   wss 3280  cuni 3975  ciun 4053   cdm 4837   cres 4839   wfn 5408  cfv 5413  cpred 25381 This theorem is referenced by:  wfrlem10  25479  wfrlem14  25483  wfrlem15  25484 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-pred 25382
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