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Theorem wfr1 7958
 Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr1.1 𝑅 We 𝐴
wfr1.2 𝑅 Se 𝐴
wfr1.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfr1 𝐹 Fn 𝐴

Proof of Theorem wfr1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 wfr1.1 . . 3 𝑅 We 𝐴
2 wfr1.2 . . 3 𝑅 Se 𝐴
3 wfr1.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
41, 2, 3wfrfun 7950 . 2 Fun 𝐹
5 eqid 2798 . . 3 (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
61, 2, 3, 5wfrlem16 7955 . 2 dom 𝐹 = 𝐴
7 df-fn 6327 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
84, 6, 7mpbir2an 710 1 𝐹 Fn 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3879  {csn 4525  ⟨cop 4531   Se wse 5476   We wwe 5477  dom cdm 5519   ↾ cres 5521  Predcpred 6115  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  wrecscwrecs 7931 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-wrecs 7932 This theorem is referenced by:  wfr3  7960  tfr1ALT  8021  bpolylem  15396
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