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Theorem frr2 33294
 Description: Law of general founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
Hypothesis
Ref Expression
frr.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frr2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem frr2
Dummy variables 𝑥 𝑦 𝑢 𝑣 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frr.1 . . . . . 6 𝐹 = frecs(𝑅, 𝐴, 𝐺)
21frr1 33293 . . . . 5 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
32fndmd 6430 . . . 4 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
43eleq2d 2875 . . 3 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹𝑋𝐴))
54pm5.32i 578 . 2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) ↔ ((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴))
6 fveq2 6650 . . . . . 6 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
7 id 22 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
8 predeq3 6123 . . . . . . . 8 (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
98reseq2d 5819 . . . . . . 7 (𝑦 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
107, 9oveq12d 7158 . . . . . 6 (𝑦 = 𝑋 → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
116, 10eqeq12d 2814 . . . . 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 Se 𝐴) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))))
13 eqid 2798 . . . . . . 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 33272 . . . . . 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, 1frrlem15 33291 . . . . . 6 (((𝑅 Fr 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 33281 . . . . 5 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑦 ∈ dom 𝐹) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1716expcom 417 . . . 4 (𝑦 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1812, 17vtoclga 3522 . . 3 (𝑋 ∈ dom 𝐹 → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
1918impcom 411 . 2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
205, 19sylbir 238 1 (((𝑅 Fr 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 2776  ∀wral 3106   ⊆ wss 3881   Fr wfr 5476   Se wse 5477  dom cdm 5520   ↾ cres 5522  Predcpred 6118   Fn wfn 6322  ‘cfv 6327  (class class class)co 7140  frecscfrecs 33266 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 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448  ax-inf2 9095 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-om 7568  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-trpred 33206  df-frecs 33267 This theorem is referenced by:  frr3  33295
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