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Theorem wfr3 7718
Description: The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in wfr1 7716 and wfr2 7717 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr3.1 𝑅 We 𝐴
wfr3.2 𝑅 Se 𝐴
wfr3.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfr3 ((𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝐺   𝑧,𝐻   𝑧,𝑅

Proof of Theorem wfr3
StepHypRef Expression
1 wfr3.1 . . 3 𝑅 We 𝐴
2 wfr3.2 . . 3 𝑅 Se 𝐴
31, 2pm3.2i 464 . 2 (𝑅 We 𝐴𝑅 Se 𝐴)
4 wfr3.3 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
51, 2, 4wfr1 7716 . . 3 𝐹 Fn 𝐴
61, 2, 4wfr2 7717 . . . 4 (𝑧𝐴 → (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
76rgen 3104 . . 3 𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85, 7pm3.2i 464 . 2 (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9 wfr3g 7695 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
103, 8, 9mp3an12 1524 1 ((𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wral 3090   Se wse 5312   We wwe 5313  cres 5357  Predcpred 5932   Fn wfn 6130  cfv 6135  wrecscwrecs 7688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-wrecs 7689
This theorem is referenced by:  tfr3ALT  7781
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