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Theorem fpr2 33401
 Description: Law of founded partial recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
Hypothesis
Ref Expression
fprr.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
fpr2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem fpr2
Dummy variables 𝑥 𝑦 𝑢 𝑣 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprr.1 . . . . . 6 𝐹 = frecs(𝑅, 𝐴, 𝐺)
21fpr1 33400 . . . . 5 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
32fndmd 6438 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
43eleq2d 2837 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹𝑋𝐴))
54biimpar 481 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐹)
6 fveq2 6658 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
7 id 22 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
8 predeq3 6130 . . . . . . . 8 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
98reseq2d 5823 . . . . . . 7 (𝑦 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
107, 9oveq12d 7168 . . . . . 6 (𝑦 = 𝑋 → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
116, 10eqeq12d 2774 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
1211imbi2d 344 . . . 4 (𝑦 = 𝑋 → (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))))
13 eqid 2758 . . . . . . 7 {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} = {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))}
1413frrlem1 33384 . . . . . 6 {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1514, 1fprlem1 33398 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝑔 ∈ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} ∧ ∈ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))})) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1614, 1, 15frrlem10 33393 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1716expcom 417 . . . 4 (𝑦 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1812, 17vtoclga 3492 . . 3 (𝑋 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
1918impcom 411 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
205, 19syldan 594 1 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070   ⊆ wss 3858   Po wpo 5441   Fr wfr 5480   Se wse 5481  dom cdm 5524   ↾ cres 5526  Predcpred 6125   Fn wfn 6330  ‘cfv 6335  (class class class)co 7150  frecscfrecs 33378 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-fr 5483  df-se 5484  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-frecs 33379 This theorem is referenced by:  fpr3  33402  norecov  33651  norec2ov  33661
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