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

Theorem wfr1 7320
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 7312 . 2 Fun 𝐹
5 eqid 2610 . . 3 (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
61, 2, 3, 5wfrlem16 7317 . 2 dom 𝐹 = 𝐴
7 df-fn 5807 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
84, 6, 7mpbir2an 957 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cun 3538  {csn 4125  cop 4131   Se wse 4995   We wwe 4996  dom cdm 5038  cres 5040  Predcpred 5596  Fun wfun 5798   Fn wfn 5799  cfv 5804  wrecscwrecs 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294
This theorem is referenced by:  wfr3  7322  tfr1ALT  7383  bpolylem  14618
  Copyright terms: Public domain W3C validator