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Theorem wfr3 7953
 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 7951 and wfr2 7952 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 473 . 2 (𝑅 We 𝐴𝑅 Se 𝐴)
4 wfr3.3 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
51, 2, 4wfr1 7951 . . 3 𝐹 Fn 𝐴
61, 2, 4wfr2 7952 . . . 4 (𝑧𝐴 → (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
76rgen 3135 . . 3 𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85, 7pm3.2i 473 . 2 (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9 wfr3g 7931 . 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 1447 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 398   = wceq 1537  ∀wral 3125   Se wse 5488   We wwe 5489   ↾ cres 5533  Predcpred 6123   Fn wfn 6326  ‘cfv 6331  wrecscwrecs 7924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-wrecs 7925 This theorem is referenced by:  tfr3ALT  8016
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