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Theorem fpr3 8329
Description: Law of well-founded recursion over a partial order, part three. 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 fpr1 8327 and fpr2 8328 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.)
Hypothesis
Ref Expression
fprr.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
fpr3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
Distinct variable groups:   𝑧,𝐹   𝑧,𝑅   𝑧,𝐴   𝑧,𝐺   𝑧,𝐻

Proof of Theorem fpr3
StepHypRef Expression
1 simpl 482 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴))
2 fprr.1 . . . . 5 𝐹 = frecs(𝑅, 𝐴, 𝐺)
32fpr1 8327 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
42fpr2 8328 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → (𝐹𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
54ralrimiva 3144 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (𝐹𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
63, 5jca 511 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
76adantr 480 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
8 simpr 484 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))))
9 fpr3g 8309 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
101, 7, 8, 9syl3anc 1370 1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wral 3059   Po wpo 5595   Fr wfr 5638   Se wse 5639  cres 5691  Predcpred 6322   Fn wfn 6558  cfv 6563  (class class class)co 7431  frecscfrecs 8304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-frecs 8305
This theorem is referenced by: (None)
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