MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frrlem15 Structured version   Visualization version   GIF version

Theorem frrlem15 9693
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 3449 . . . . . 6 𝑥 ∈ V
2 vex 3449 . . . . . 6 𝑢 ∈ V
31, 2breldm 5864 . . . . 5 (𝑥𝑔𝑢𝑥 ∈ dom 𝑔)
43adantr 481 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom 𝑔)
5 vex 3449 . . . . . 6 𝑣 ∈ V
61, 5breldm 5864 . . . . 5 (𝑥𝑣𝑥 ∈ dom )
76adantl 482 . . . 4 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ dom )
84, 7elind 4154 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → 𝑥 ∈ (dom 𝑔 ∩ dom ))
9 id 22 . . 3 ((𝑥𝑔𝑢𝑥𝑣) → (𝑥𝑔𝑢𝑥𝑣))
102brresi 5946 . . . . 5 (𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢))
115brresi 5946 . . . . 5 (𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣 ↔ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣))
1210, 11anbi12i 627 . . . 4 ((𝑥(𝑔 ↾ (dom 𝑔 ∩ dom ))𝑢𝑥( ↾ (dom 𝑔 ∩ dom ))𝑣) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)))
13 an4 654 . . . 4 (((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑔𝑢) ∧ (𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥𝑣)) ↔ ((𝑥 ∈ (dom 𝑔 ∩ dom ) ∧ 𝑥 ∈ (dom 𝑔 ∩ dom )) ∧ (𝑥𝑔𝑢𝑥𝑣)))
1412, 13bitri 274 . . 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 4188 . . . . . . . . 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 8219 . . . . . . . . 9 (𝑔𝐵 → dom 𝑔𝐴)
1916, 18sstrid 3955 . . . . . . . 8 (𝑔𝐵 → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
2019ad2antrl 726 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (dom 𝑔 ∩ dom ) ⊆ 𝐴)
21 simpll 765 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Fr 𝐴)
22 frss 5600 . . . . . . 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 767 . . . . . . 7 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → 𝑅 Se 𝐴)
25 sess2 5602 . . . . . . 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 8220 . . . . . . 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 482 . . . . . 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 8220 . . . . . . . . 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 4161 . . . . . . . . . . . 12 (dom 𝑔 ∩ dom ) = (dom ∩ dom 𝑔)
3130reseq2i 5934 . . . . . . . . . . 11 ( ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom ∩ dom 𝑔))
32 fneq12 6598 . . . . . . . . . . 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 690 . . . . . . . . . 10 (( ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ↔ ( ↾ (dom ∩ dom 𝑔)) Fn (dom ∩ dom 𝑔))
3431fveq1i 6843 . . . . . . . . . . . 12 (( ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (( ↾ (dom ∩ dom 𝑔))‘𝑎)
35 predeq2 6256 . . . . . . . . . . . . . . 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 5935 . . . . . . . . . . . . 13 (( ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)) = (( ↾ (dom ∩ dom 𝑔)) ↾ Pred(𝑅, (dom ∩ dom 𝑔), 𝑎))
3837oveq2i 7368 . . . . . . . . . . . 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 3315 . . . . . . . . . 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 627 . . . . . . . . 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 459 . . . . . . 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 482 . . . . . 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 9692 . . . . . 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 1376 . . . . 5 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
4746breqd 5116 . . . 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 8218 . . . . . 6 (𝑔𝐵 → Fun 𝑔)
5049funresd 6544 . . . . 5 (𝑔𝐵 → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
5150ad2antrl 726 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → Fun (𝑔 ↾ (dom 𝑔 ∩ dom )))
52 dffun2 6506 . . . . 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 2179 . . . . . 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 2178 . . . . 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 505 . . . 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 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  cin 3909  wss 3910   class class class wbr 5105   Fr wfr 5585   Se wse 5586  dom cdm 5633  cres 5635  Rel wrel 5638  Predcpred 6252  Fun wfun 6490   Fn wfn 6491  cfv 6496  (class class class)co 7357  frecscfrecs 8211
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-ttrcl 9644
This theorem is referenced by:  frr1  9695  frr2  9696
  Copyright terms: Public domain W3C validator