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Theorem wfrlem13 28932
 Description: Lemma for well-founded recursion. From here through wfrlem16 28935, we aim to prove that . We do this by supposing that there is an element of that is not in . We then define by extending with the appropriate value at . We then show that cannot be an minimal element of , meaning that must be empty, so . Here, we show that is a function extending the domain of by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem13.1
wfrlem13.2 Se
wfrlem13.3 wrecs
wfrlem13.4
Assertion
Ref Expression
wfrlem13
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem wfrlem13
StepHypRef Expression
1 wfrlem13.1 . . . . . 6
2 wfrlem13.2 . . . . . 6 Se
3 wfrlem13.3 . . . . . 6 wrecs
41, 2, 3wfrlem11 28930 . . . . 5
5 vex 3116 . . . . . 6
6 fvex 5874 . . . . . 6
75, 6funsn 5634 . . . . 5
84, 7pm3.2i 455 . . . 4
96dmsnop 5480 . . . . . 6
109ineq2i 3697 . . . . 5
11 eldifn 3627 . . . . . 6
12 disjsn 4088 . . . . . 6
1311, 12sylibr 212 . . . . 5
1410, 13syl5eq 2520 . . . 4
15 funun 5628 . . . 4
168, 14, 15sylancr 663 . . 3
17 dmun 5207 . . . 4
189uneq2i 3655 . . . 4
1917, 18eqtri 2496 . . 3
2016, 19jctir 538 . 2
21 wfrlem13.4 . . . 4
2221fneq1i 5673 . . 3
23 df-fn 5589 . . 3
2422, 23bitri 249 . 2
2520, 24sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   wceq 1379   wcel 1767   cdif 3473   cun 3474   cin 3475  c0 3785  csn 4027  cop 4033   Se wse 4836   wwe 4837   cdm 4999   cres 5001   wfun 5580   wfn 5581  cfv 5586  cpred 28820  wrecscwrecs 28912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-pred 28821  df-wrecs 28913 This theorem is referenced by:  wfrlem14  28933  wfrlem15  28934
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