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Theorem frrlem15 9798
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 3483 . . . . . 6 𝑥 ∈ V
2 vex 3483 . . . . . 6 𝑢 ∈ V
31, 2breldm 5918 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
43adantr 480 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom 𝑔)
5 vex 3483 . . . . . 6 𝑣 ∈ V
61, 5breldm 5918 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
76adantl 481 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom )
84, 7elind 4199 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 id 22 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥𝑔𝑢𝑥𝑣))
102brresi 6005 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢))
115brresi 6005 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣))
1210, 11anbi12i 628 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
13 an4 656 . . . 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 1344 . 2 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
16 inss1 4236 . . . . . . . . 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 8314 . . . . . . . . 9 (𝑔𝐵 → dom 𝑔𝐴)
1916, 18sstrid 3994 . . . . . . . 8 (𝑔𝐵 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2019ad2antrl 728 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
21 simpll 766 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Fr 𝐴)
22 frss 5648 . . . . . . 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 768 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Se 𝐴)
25 sess2 5650 . . . . . . 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 8315 . . . . . . 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 8315 . . . . . . . . 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 4208 . . . . . . . . . . . 12 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5993 . . . . . . . . . . 11 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
32 fneq12 6663 . . . . . . . . . . 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 692 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3431fveq1i 6906 . . . . . . . . . . . 12 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
35 predeq2 6323 . . . . . . . . . . . . . . 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 5994 . . . . . . . . . . . . 13 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3837oveq2i 7443 . . . . . . . . . . . 12 (𝑎𝐺(( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎))) = (𝑎𝐺(( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎)))
3934, 38eqeq12i 2754 . . . . . . . . . . 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 3343 . . . . . . . . . 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 628 . . . . . . . . 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 234 . . . . . . . 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 9797 . . . . . 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 1377 . . . . 5 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4746breqd 5153 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣))
4847biimprd 248 . . 3 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑣))
4917frrlem2 8313 . . . . . 6 (𝑔𝐵 → Fun 𝑔)
5049funresd 6608 . . . . 5 (𝑔𝐵 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
5150ad2antrl 728 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
52 dffun2 6570 . . . . 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 2185 . . . . . 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 2184 . . . . 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 605 . 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 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  cin 3949  wss 3950   class class class wbr 5142   Fr wfr 5633   Se wse 5634  dom cdm 5684  cres 5686  Rel wrel 5689  Predcpred 6319  Fun wfun 6554   Fn wfn 6555  cfv 6560  (class class class)co 7432  frecscfrecs 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-ttrcl 9749
This theorem is referenced by:  frr1  9800  frr2  9801
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