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Theorem fpr2 32751
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 32750 . . . . 5 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
32fndmd 6333 . . . 4 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
43eleq2d 2870 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹𝑋𝐴))
54biimpar 478 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐹)
6 fveq2 6545 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
7 id 22 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
8 predeq3 6034 . . . . . . . 8 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
98reseq2d 5741 . . . . . . 7 (𝑦 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
107, 9oveq12d 7041 . . . . . 6 (𝑦 = 𝑋 → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
116, 10eqeq12d 2812 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
1211imbi2d 342 . . . 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 2797 . . . . . . 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 32734 . . . . . 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 32748 . . . . . 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 32743 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1716expcom 414 . . . 4 (𝑦 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1812, 17vtoclga 3519 . . 3 (𝑋 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
1918impcom 408 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
205, 19syldan 591 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 396  w3a 1080   = wceq 1525  wex 1765  wcel 2083  {cab 2777  wral 3107  wss 3865   Po wpo 5367   Fr wfr 5406   Se wse 5407  dom cdm 5450  cres 5452  Predcpred 6029   Fn wfn 6227  cfv 6232  (class class class)co 7023  frecscfrecs 32728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-po 5369  df-fr 5409  df-se 5410  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-frecs 32729
This theorem is referenced by:  fpr3  32752
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