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Theorem wfr2 8332
Description: The Principle of Well-Ordered Recursion, part 2 of 3. Next, we show that the value of 𝐹 at any 𝑋𝐴 is 𝐺 applied to all "previous" values of 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
wfr2.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfr2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2
StepHypRef Expression
1 wfr2.1 . . . . . 6 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
21wfr1 8331 . . . . 5 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
32fndmd 6645 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
43eleq2d 2811 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹𝑋𝐴))
54biimpar 477 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐹)
61wfr2a 8330 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
75, 6syldan 590 1 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098   Se wse 5620   We wwe 5621  dom cdm 5667  cres 5669  Predcpred 6290  cfv 6534  wrecscwrecs 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-2nd 7970  df-frecs 8262  df-wrecs 8293
This theorem is referenced by:  wfr3  8333  wfr3OLD  8334  tfr2ALT  8397  bpolylem  15990
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