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Theorem wfr1 7059
 Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function . Then, using a base class and a well-ordering of , we define a function . This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of . We begin by showing that is a function over . (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr1.1
wfr1.2 Se
wfr1.3 wrecs
Assertion
Ref Expression
wfr1

Proof of Theorem wfr1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 wfr1.1 . . 3
2 wfr1.2 . . 3 Se
3 wfr1.3 . . 3 wrecs
41, 2, 3wfrfun 7051 . 2
5 eqid 2422 . . 3
61, 2, 3, 5wfrlem16 7056 . 2
7 df-fn 5601 . 2
84, 6, 7mpbir2an 928 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437   cun 3434  csn 3996  cop 4002   Se wse 4807   wwe 4808   cdm 4850   cres 4852  cpred 5395   wfun 5592   wfn 5593  cfv 5598  wrecscwrecs 7032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594 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 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-wrecs 7033 This theorem is referenced by:  wfr3  7061  tfr1ALT  7123  bpolylem  14089
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