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Theorem wfrlem13 7059
 Description: Lemma for well-founded recursion. From here through wfrlem16 7062, 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, 3wfrfun 7057 . . . . 5
5 vex 3083 . . . . . 6
6 fvex 5891 . . . . . 6
75, 6funsn 5649 . . . . 5
84, 7pm3.2i 456 . . . 4
96dmsnop 5329 . . . . . 6
109ineq2i 3661 . . . . 5
11 eldifn 3588 . . . . . 6
12 disjsn 4060 . . . . . 6
1311, 12sylibr 215 . . . . 5
1410, 13syl5eq 2475 . . . 4
15 funun 5643 . . . 4
168, 14, 15sylancr 667 . . 3
17 dmun 5060 . . . 4
189uneq2i 3617 . . . 4
1917, 18eqtri 2451 . . 3
2016, 19jctir 540 . 2
21 wfrlem13.4 . . . 4
2221fneq1i 5688 . . 3
23 df-fn 5604 . . 3
2422, 23bitri 252 . 2
2520, 24sylibr 215 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 370   wceq 1437   wcel 1872   cdif 3433   cun 3434   cin 3435  c0 3761  csn 3998  cop 4004   Se wse 4810   wwe 4811   cdm 4853   cres 4855  cpred 5398   wfun 5595   wfn 5596  cfv 5601  wrecscwrecs 7038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-wrecs 7039 This theorem is referenced by:  wfrlem14  7060  wfrlem15  7061
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