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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
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