Metamath Proof Explorer

Theorem frrlem12

Description: Lemma for well-founded recursion. Next, we calculate the value of C . (Contributed by Scott Fenton, 7-Dec-2022)

Ref Expression
Hypotheses frrlem11.1 
|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) }
frrlem11.2 
|- F = frecs ( R , A , G )
frrlem11.3 
|- ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) )
frrlem11.4 
|- C = ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } )
frrlem12.5 
|- ( ph -> R Fr A )
frrlem12.6 
|- ( ( ph /\ z e. A ) -> Pred ( R , A , z ) C_ S )
frrlem12.7 
|- ( ( ph /\ z e. A ) -> A. w e. S Pred ( R , A , w ) C_ S )
Assertion frrlem12 
|- ( ( ph /\ z e. ( A \ dom F ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) )

Proof

Step Hyp Ref Expression
1 frrlem11.1 
 |-  B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) }
2 frrlem11.2 
 |-  F = frecs ( R , A , G )
3 frrlem11.3 
 |-  ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) )
4 frrlem11.4 
 |-  C = ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } )
5 frrlem12.5 
 |-  ( ph -> R Fr A )
6 frrlem12.6 
 |-  ( ( ph /\ z e. A ) -> Pred ( R , A , z ) C_ S )
7 frrlem12.7 
 |-  ( ( ph /\ z e. A ) -> A. w e. S Pred ( R , A , w ) C_ S )
8 elun 
 |-  ( w e. ( ( S i^i dom F ) u. { z } ) <-> ( w e. ( S i^i dom F ) \/ w e. { z } ) )
9 velsn 
 |-  ( w e. { z } <-> w = z )
10 9 orbi2i 
 |-  ( ( w e. ( S i^i dom F ) \/ w e. { z } ) <-> ( w e. ( S i^i dom F ) \/ w = z ) )
11 8 10 bitri 
 |-  ( w e. ( ( S i^i dom F ) u. { z } ) <-> ( w e. ( S i^i dom F ) \/ w = z ) )
12 elinel2 
 |-  ( w e. ( S i^i dom F ) -> w e. dom F )
13 1 frrlem1 
 |-  B = { p | E. q ( p Fn q /\ ( q C_ A /\ A. w e. q Pred ( R , A , w ) C_ q ) /\ A. w e. q ( p ` w ) = ( w G ( p |` Pred ( R , A , w ) ) ) ) }
14 breq1 
 |-  ( x = q -> ( x g u <-> q g u ) )
15 breq1 
 |-  ( x = q -> ( x h v <-> q h v ) )
16 14 15 anbi12d 
 |-  ( x = q -> ( ( x g u /\ x h v ) <-> ( q g u /\ q h v ) ) )
17 16 imbi1d 
 |-  ( x = q -> ( ( ( x g u /\ x h v ) -> u = v ) <-> ( ( q g u /\ q h v ) -> u = v ) ) )
18 17 imbi2d 
 |-  ( x = q -> ( ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) <-> ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( q g u /\ q h v ) -> u = v ) ) ) )
19 18 3 chvarvv 
 |-  ( ( ph /\ ( g e. B /\ h e. B ) ) -> ( ( q g u /\ q h v ) -> u = v ) )
20 13 2 19 frrlem10 
 |-  ( ( ph /\ w e. dom F ) -> ( F ` w ) = ( w G ( F |` Pred ( R , A , w ) ) ) )
21 12 20 sylan2 
 |-  ( ( ph /\ w e. ( S i^i dom F ) ) -> ( F ` w ) = ( w G ( F |` Pred ( R , A , w ) ) ) )
22 21 adantlr 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( F ` w ) = ( w G ( F |` Pred ( R , A , w ) ) ) )
23 4 fveq1i 
 |-  ( C ` w ) = ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` w )
24 1 2 3 frrlem9 
 |-  ( ph -> Fun F )
25 24 funresd 
 |-  ( ph -> Fun ( F |` S ) )
26 dmres 
 |-  dom ( F |` S ) = ( S i^i dom F )
27 df-fn 
 |-  ( ( F |` S ) Fn ( S i^i dom F ) <-> ( Fun ( F |` S ) /\ dom ( F |` S ) = ( S i^i dom F ) ) )
28 25 26 27 sylanblrc 
 |-  ( ph -> ( F |` S ) Fn ( S i^i dom F ) )
29 28 adantr 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( F |` S ) Fn ( S i^i dom F ) )
30 29 adantr 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( F |` S ) Fn ( S i^i dom F ) )
31 vex 
 |-  z e. _V
32 ovex 
 |-  ( z G ( F |` Pred ( R , A , z ) ) ) e. _V
33 31 32 fnsn 
 |-  { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } Fn { z }
34 33 a1i 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } Fn { z } )
35 eldifn 
 |-  ( z e. ( A \ dom F ) -> -. z e. dom F )
36 elinel2 
 |-  ( z e. ( S i^i dom F ) -> z e. dom F )
37 35 36 nsyl 
 |-  ( z e. ( A \ dom F ) -> -. z e. ( S i^i dom F ) )
38 disjsn 
 |-  ( ( ( S i^i dom F ) i^i { z } ) = (/) <-> -. z e. ( S i^i dom F ) )
39 37 38 sylibr 
 |-  ( z e. ( A \ dom F ) -> ( ( S i^i dom F ) i^i { z } ) = (/) )
40 39 adantl 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( S i^i dom F ) i^i { z } ) = (/) )
41 40 adantr 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( ( S i^i dom F ) i^i { z } ) = (/) )
42 simpr 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> w e. ( S i^i dom F ) )
43 fvun1 
 |-  ( ( ( F |` S ) Fn ( S i^i dom F ) /\ { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } Fn { z } /\ ( ( ( S i^i dom F ) i^i { z } ) = (/) /\ w e. ( S i^i dom F ) ) ) -> ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` w ) = ( ( F |` S ) ` w ) )
44 30 34 41 42 43 syl112anc 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` w ) = ( ( F |` S ) ` w ) )
45 23 44 eqtrid 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( C ` w ) = ( ( F |` S ) ` w ) )
46 elinel1 
 |-  ( w e. ( S i^i dom F ) -> w e. S )
47 46 adantl 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> w e. S )
48 47 fvresd 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( ( F |` S ) ` w ) = ( F ` w ) )
49 45 48 eqtrd 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( C ` w ) = ( F ` w ) )
50 1 2 3 4 frrlem11 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> C Fn ( ( S i^i dom F ) u. { z } ) )
51 fnfun 
 |-  ( C Fn ( ( S i^i dom F ) u. { z } ) -> Fun C )
52 50 51 syl 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> Fun C )
53 52 adantr 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> Fun C )
54 ssun1 
 |-  ( F |` S ) C_ ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } )
55 54 4 sseqtrri 
 |-  ( F |` S ) C_ C
56 55 a1i 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( F |` S ) C_ C )
57 eldifi 
 |-  ( z e. ( A \ dom F ) -> z e. A )
58 57 7 sylan2 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> A. w e. S Pred ( R , A , w ) C_ S )
59 rspa 
 |-  ( ( A. w e. S Pred ( R , A , w ) C_ S /\ w e. S ) -> Pred ( R , A , w ) C_ S )
60 58 46 59 syl2an 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> Pred ( R , A , w ) C_ S )
61 1 2 frrlem8 
 |-  ( w e. dom F -> Pred ( R , A , w ) C_ dom F )
62 12 61 syl 
 |-  ( w e. ( S i^i dom F ) -> Pred ( R , A , w ) C_ dom F )
63 62 adantl 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> Pred ( R , A , w ) C_ dom F )
64 60 63 ssind 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> Pred ( R , A , w ) C_ ( S i^i dom F ) )
65 64 26 sseqtrrdi 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> Pred ( R , A , w ) C_ dom ( F |` S ) )
66 fun2ssres 
 |-  ( ( Fun C /\ ( F |` S ) C_ C /\ Pred ( R , A , w ) C_ dom ( F |` S ) ) -> ( C |` Pred ( R , A , w ) ) = ( ( F |` S ) |` Pred ( R , A , w ) ) )
67 53 56 65 66 syl3anc 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( C |` Pred ( R , A , w ) ) = ( ( F |` S ) |` Pred ( R , A , w ) ) )
68 60 resabs1d 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( ( F |` S ) |` Pred ( R , A , w ) ) = ( F |` Pred ( R , A , w ) ) )
69 67 68 eqtrd 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( C |` Pred ( R , A , w ) ) = ( F |` Pred ( R , A , w ) ) )
70 69 oveq2d 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( w G ( C |` Pred ( R , A , w ) ) ) = ( w G ( F |` Pred ( R , A , w ) ) ) )
71 22 49 70 3eqtr4d 
 |-  ( ( ( ph /\ z e. ( A \ dom F ) ) /\ w e. ( S i^i dom F ) ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) )
72 71 ex 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( w e. ( S i^i dom F ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) )
73 31 32 fvsn 
 |-  ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ` z ) = ( z G ( F |` Pred ( R , A , z ) ) )
74 4 fveq1i 
 |-  ( C ` z ) = ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` z )
75 33 a1i 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } Fn { z } )
76 vsnid 
 |-  z e. { z }
77 76 a1i 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> z e. { z } )
78 fvun2 
 |-  ( ( ( F |` S ) Fn ( S i^i dom F ) /\ { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } Fn { z } /\ ( ( ( S i^i dom F ) i^i { z } ) = (/) /\ z e. { z } ) ) -> ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` z ) = ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ` z ) )
79 29 75 40 77 78 syl112anc 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) ` z ) = ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ` z ) )
80 74 79 eqtrid 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( C ` z ) = ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ` z ) )
81 4 reseq1i 
 |-  ( C |` Pred ( R , A , z ) ) = ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) |` Pred ( R , A , z ) )
82 resundir 
 |-  ( ( ( F |` S ) u. { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) |` Pred ( R , A , z ) ) = ( ( ( F |` S ) |` Pred ( R , A , z ) ) u. ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) )
83 81 82 eqtri 
 |-  ( C |` Pred ( R , A , z ) ) = ( ( ( F |` S ) |` Pred ( R , A , z ) ) u. ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) )
84 57 6 sylan2 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> Pred ( R , A , z ) C_ S )
85 84 resabs1d 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( F |` S ) |` Pred ( R , A , z ) ) = ( F |` Pred ( R , A , z ) ) )
86 predfrirr 
 |-  ( R Fr A -> -. z e. Pred ( R , A , z ) )
87 5 86 syl 
 |-  ( ph -> -. z e. Pred ( R , A , z ) )
88 87 adantr 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> -. z e. Pred ( R , A , z ) )
89 ressnop0 
 |-  ( -. z e. Pred ( R , A , z ) -> ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) = (/) )
90 88 89 syl 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) = (/) )
91 85 90 uneq12d 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( ( F |` S ) |` Pred ( R , A , z ) ) u. ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) ) = ( ( F |` Pred ( R , A , z ) ) u. (/) ) )
92 un0 
 |-  ( ( F |` Pred ( R , A , z ) ) u. (/) ) = ( F |` Pred ( R , A , z ) )
93 91 92 eqtrdi 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( ( F |` S ) |` Pred ( R , A , z ) ) u. ( { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } |` Pred ( R , A , z ) ) ) = ( F |` Pred ( R , A , z ) ) )
94 83 93 eqtrid 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( C |` Pred ( R , A , z ) ) = ( F |` Pred ( R , A , z ) ) )
95 94 oveq2d 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( z G ( C |` Pred ( R , A , z ) ) ) = ( z G ( F |` Pred ( R , A , z ) ) ) )
96 73 80 95 3eqtr4a 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( C ` z ) = ( z G ( C |` Pred ( R , A , z ) ) ) )
97 fveq2 
 |-  ( w = z -> ( C ` w ) = ( C ` z ) )
98 id 
 |-  ( w = z -> w = z )
99 predeq3 
 |-  ( w = z -> Pred ( R , A , w ) = Pred ( R , A , z ) )
100 99 reseq2d 
 |-  ( w = z -> ( C |` Pred ( R , A , w ) ) = ( C |` Pred ( R , A , z ) ) )
101 98 100 oveq12d 
 |-  ( w = z -> ( w G ( C |` Pred ( R , A , w ) ) ) = ( z G ( C |` Pred ( R , A , z ) ) ) )
102 97 101 eqeq12d 
 |-  ( w = z -> ( ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) <-> ( C ` z ) = ( z G ( C |` Pred ( R , A , z ) ) ) ) )
103 96 102 syl5ibrcom 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( w = z -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) )
104 72 103 jaod 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( ( w e. ( S i^i dom F ) \/ w = z ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) )
105 11 104 syl5bi 
 |-  ( ( ph /\ z e. ( A \ dom F ) ) -> ( w e. ( ( S i^i dom F ) u. { z } ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) ) )
106 105 3impia 
 |-  ( ( ph /\ z e. ( A \ dom F ) /\ w e. ( ( S i^i dom F ) u. { z } ) ) -> ( C ` w ) = ( w G ( C |` Pred ( R , A , w ) ) ) )