Theorem wfr1 7967

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.

Step	Hyp	Ref	Expression
1		wfr1.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfr1.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfr1.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4	1, 2, 3	wfrfun 7959	. 2 ⊢ Fun 𝐹
5		eqid 2821	. . 3 ⊢ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
6	1, 2, 3, 5	wfrlem16 7964	. 2 ⊢ dom 𝐹 = 𝐴
7		df-fn 6353	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
8	4, 6, 7	mpbir2an 709	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1533   ∪ cun 3934  {csn 4561  ⟨cop 4567   Se wse 5507   We wwe 5508  dom cdm 5550   ↾ cres 5552  Predcpred 6142  Fun wfun 6344   Fn wfn 6345  ‘cfv 6350  wrecscwrecs 7940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-wrecs 7941

This theorem is referenced by:  wfr3  7969  tfr1ALT  8030  bpolylem  15396