Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfr2 Structured version   Visualization version   GIF version

Theorem wfr2 7961
 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 (𝑋𝐴 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef 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 2801 . . . 4 (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩}) = (𝐹 ∪ {⟨𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))⟩})
51, 2, 3, 4wfrlem16 7957 . . 3 dom 𝐹 = 𝐴
65eleq2i 2884 . 2 (𝑋 ∈ dom 𝐹𝑋𝐴)
71, 2, 3wfr2a 7959 . 2 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
86, 7sylbir 238 1 (𝑋𝐴 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112   ∪ cun 3882  {csn 4528  ⟨cop 4534   Se wse 5480   We wwe 5481  dom cdm 5523   ↾ cres 5525  Predcpred 6119  ‘cfv 6328  wrecscwrecs 7933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-wrecs 7934 This theorem is referenced by:  wfr3  7962  tfr2ALT  8024  bpolylem  15397
 Copyright terms: Public domain W3C validator