Theorem wfr2 7700

Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of 𝐹 at any 𝑋 ∈ 𝐴 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfr2.1	⊢ 𝑅 We 𝐴
wfr2.2	⊢ 𝑅 Se 𝐴
wfr2.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr2	⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfr2.1	. . . 4 ⊢ 𝑅 We 𝐴
2		wfr2.2	. . . 4 ⊢ 𝑅 Se 𝐴
3		wfr2.3	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		eqid 2825	. . . 4 ⊢ (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩}) = (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩})
5	1, 2, 3, 4	wfrlem16 7696	. . 3 ⊢ dom 𝐹 = 𝐴
6	5	eleq2i 2898	. 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴)
7	1, 2, 3	wfr2a 7698	. 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
8	6, 7	sylbir 227	1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166   ∪ cun 3796  {csn 4397  ⟨cop 4403   Se wse 5299   We wwe 5300  dom cdm 5342   ↾ cres 5344  Predcpred 5919  ‘cfv 6123  wrecscwrecs 7671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-wrecs 7672

This theorem is referenced by:  wfr3  7701  tfr2ALT  7763  bpolylem  15151