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Theorem frrlem15 9747
Description: Lemma for general well-founded recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)
Hypotheses
Ref Expression
frrlem15.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem15.2 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrlem15 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑔,,𝑢,𝑣   𝑅,𝑓,𝑥,𝑦,𝑔,,𝑢,𝑣   𝑓,𝐺,𝑥,𝑦,𝑔,,𝑢,𝑣
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔,)   𝐹(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔,)

Proof of Theorem frrlem15
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 vex 3470 . . . . . 6 𝑥 ∈ V
2 vex 3470 . . . . . 6 𝑢 ∈ V
31, 2breldm 5898 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
43adantr 480 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom 𝑔)
5 vex 3470 . . . . . 6 𝑣 ∈ V
61, 5breldm 5898 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
76adantl 481 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom )
84, 7elind 4186 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 id 22 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥𝑔𝑢𝑥𝑣))
102brresi 5980 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢))
115brresi 5980 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣))
1210, 11anbi12i 626 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
13 an4 653 . . . 4 (((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑔𝑢𝑥𝑣)))
1412, 13bitri 275 . . 3 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑔𝑢𝑥𝑣)))
158, 8, 9, 14syl21anbrc 1341 . 2 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
16 inss1 4220 . . . . . . . . 9 (dom 𝑔 ∩ dom ) ⊆ dom 𝑔
17 frrlem15.1 . . . . . . . . . 10 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1817frrlem3 8268 . . . . . . . . 9 (𝑔𝐵 → dom 𝑔𝐴)
1916, 18sstrid 3985 . . . . . . . 8 (𝑔𝐵 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2019ad2antrl 725 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
21 simpll 764 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Fr 𝐴)
22 frss 5633 . . . . . . 7 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 Fr 𝐴𝑅 Fr (dom 𝑔 ∩ dom )))
2320, 21, 22sylc 65 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Fr (dom 𝑔 ∩ dom ))
24 simplr 766 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Se 𝐴)
25 sess2 5635 . . . . . . 7 ((dom 𝑔 ∩ dom ) ⊆ 𝐴 → (𝑅 Se 𝐴𝑅 Se (dom 𝑔 ∩ dom )))
2620, 24, 25sylc 65 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Se (dom 𝑔 ∩ dom ))
2717frrlem4 8269 . . . . . . 7 ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
2827adantl 481 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
2917frrlem4 8269 . . . . . . . . 9 ((𝐵𝑔𝐵) → (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
30 incom 4193 . . . . . . . . . . . 12 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5968 . . . . . . . . . . 11 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
32 fneq12 6635 . . . . . . . . . . 11 ((( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔)) ∧ (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔)))
3331, 30, 32mp2an 689 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3431fveq1i 6882 . . . . . . . . . . . 12 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
35 predeq2 6293 . . . . . . . . . . . . . . 15 ((dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔) → Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3630, 35ax-mp 5 . . . . . . . . . . . . . 14 Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎) = Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)
3731, 36reseq12i 5969 . . . . . . . . . . . . 13 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3837oveq2i 7412 . . . . . . . . . . . 12 (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
3934, 38eqeq12i 2742 . . . . . . . . . . 11 ((( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ (( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4030, 39raleqbii 3330 . . . . . . . . . 10 (∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) ↔ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))))
4133, 40anbi12i 626 . . . . . . . . 9 ((( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))) ↔ (( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔) ∧ ∀𝑎 ∈ (dom ∩ dom 𝑔)(( ↾ (dom ∩ dom 𝑔))‘𝑎) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))))
4229, 41sylibr 233 . . . . . . . 8 ((𝐵𝑔𝐵) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
4342ancoms 458 . . . . . . 7 ((𝑔𝐵𝐵) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
4443adantl 481 . . . . . 6 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))
45 frr3g 9746 . . . . . 6 (((𝑅 Fr (dom 𝑔 ∩ dom ) ∧ 𝑅 Se (dom 𝑔 ∩ dom )) ∧ ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))) ∧ (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )(( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))))) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4623, 26, 28, 44, 45syl211anc 1373 . . . . 5 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4746breqd 5149 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
4847biimprd 247 . . 3 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣))
4917frrlem2 8267 . . . . . 6 (𝑔𝐵 → Fun 𝑔)
5049funresd 6581 . . . . 5 (𝑔𝐵 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
5150ad2antrl 725 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
52 dffun2 6543 . . . . 5 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) ↔ (Rel (𝑔 ↾ (dom 𝑔 ∩ dom )) ∧ ∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣)))
53 2sp 2171 . . . . . 6 (∀𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5453sps 2170 . . . . 5 (∀𝑥𝑢𝑣((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5552, 54simplbiim 504 . . . 4 (Fun (𝑔 ↾ (dom 𝑔 ∩ dom )) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5651, 55syl 17 . . 3 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5748, 56sylan2d 604 . 2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) → 𝑢 = 𝑣))
5815, 57syl5 34 1 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wral 3053  cin 3939  wss 3940   class class class wbr 5138   Fr wfr 5618   Se wse 5619  dom cdm 5666  cres 5668  Rel wrel 5671  Predcpred 6289  Fun wfun 6527   Fn wfn 6528  cfv 6533  (class class class)co 7401  frecscfrecs 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-inf2 9631
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-ttrcl 9698
This theorem is referenced by:  frr1  9749  frr2  9750
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