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Theorem wfr3 7605
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 7603 and wfr2 7604 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 470 . 2 (𝑅 We 𝐴𝑅 Se 𝐴)
4 wfr3.3 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
51, 2, 4wfr1 7603 . . 3 𝐹 Fn 𝐴
61, 2, 4wfr2 7604 . . . 4 (𝑧𝐴 → (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
76rgen 3060 . . 3 𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85, 7pm3.2i 470 . 2 (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9 wfr3g 7583 . 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 1563 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 383   = wceq 1632  wral 3050   Se wse 5223   We wwe 5224  cres 5268  Predcpred 5840   Fn wfn 6044  cfv 6049  wrecscwrecs 7576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7577
This theorem is referenced by:  tfr3ALT  7668
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