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Theorem wfr2 8377
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 8376 . . . . 5 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
32fndmd 6672 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
43eleq2d 2826 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹𝑋𝐴))
54biimpar 477 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐹)
61wfr2a 8375 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
75, 6syldan 591 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 1539  wcel 2107   Se wse 5634   We wwe 5635  dom cdm 5684  cres 5686  Predcpred 6319  cfv 6560  wrecscwrecs 8337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338
This theorem is referenced by:  wfr3  8378  wfr3OLD  8379  tfr2ALT  8442  bpolylem  16085
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