MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfr1 Structured version   Visualization version   GIF version

Theorem wfr1 7771
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 7763 . 2 Fun 𝐹
5 eqid 2772 . . 3 (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
61, 2, 3, 5wfrlem16 7768 . 2 dom 𝐹 = 𝐴
7 df-fn 6185 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
84, 6, 7mpbir2an 698 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1507  cun 3821  {csn 4435  cop 4441   Se wse 5358   We wwe 5359  dom cdm 5401  cres 5403  Predcpred 5979  Fun wfun 6176   Fn wfn 6177  cfv 6182  wrecscwrecs 7743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-wrecs 7744
This theorem is referenced by:  wfr3  7773  tfr1ALT  7834  bpolylem  15256
  Copyright terms: Public domain W3C validator