Metamath Proof Explorer

Theorem wfrlem5

Description: Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypotheses wfrlem5.1 ⊢ 𝑅 We 𝐴
wfrlem5.2 ⊢ 𝑅 Se 𝐴
wfrlem5.3 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
Assertion wfrlem5 ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )

Proof

Step Hyp Ref Expression
1 wfrlem5.1 ⊢ 𝑅 We 𝐴
2 wfrlem5.2 ⊢ 𝑅 Se 𝐴
3 wfrlem5.3 ⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
4 vex ⊢ 𝑥 ∈ V
5 vex ⊢ 𝑢 ∈ V
6 4 5 breldm ⊢ ( 𝑥 𝑔 𝑢 → 𝑥 ∈ dom 𝑔 )
7 vex ⊢ 𝑣 ∈ V
8 4 7 breldm ⊢ ( 𝑥 ℎ 𝑣 → 𝑥 ∈ dom ℎ )
9 6 8 anim12i ⊢ ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → ( 𝑥 ∈ dom 𝑔 ∧ 𝑥 ∈ dom ℎ ) )
10 elin ⊢ ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ↔ ( 𝑥 ∈ dom 𝑔 ∧ 𝑥 ∈ dom ℎ ) )
11 9 10 sylibr ⊢ ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) )
12 anandi ⊢ ( ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) ) ↔ ( ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 𝑔 𝑢 ) ∧ ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 ℎ 𝑣 ) ) )
13 5 brresi ⊢ ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ↔ ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 𝑔 𝑢 ) )
14 7 brresi ⊢ ( 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ↔ ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 ℎ 𝑣 ) )
15 13 14 anbi12i ⊢ ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) ↔ ( ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 𝑔 𝑢 ) ∧ ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑥 ℎ 𝑣 ) ) )
16 12 15 sylbb2 ⊢ ( ( 𝑥 ∈ ( dom 𝑔 ∩ dom ℎ ) ∧ ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) ) → ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) )
17 11 16 mpancom ⊢ ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) )
18 3 wfrlem3 ⊢ ( 𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴 )
19 ssinss1 ⊢ ( dom 𝑔 ⊆ 𝐴 → ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 )
20 wess ⊢ ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 → ( 𝑅 We 𝐴 → 𝑅 We ( dom 𝑔 ∩ dom ℎ ) ) )
21 1 20 mpi ⊢ ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 → 𝑅 We ( dom 𝑔 ∩ dom ℎ ) )
22 sess2 ⊢ ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 → ( 𝑅 Se 𝐴 → 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) ) )
23 2 22 mpi ⊢ ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 → 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) )
24 21 23 jca ⊢ ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 → ( 𝑅 We ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) ) )
25 18 19 24 3syl ⊢ ( 𝑔 ∈ 𝐵 → ( 𝑅 We ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) ) )
26 25 adantr ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( 𝑅 We ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) ) )
27 3 wfrlem4 ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) )
28 3 wfrlem4 ⊢ ( ( ℎ ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) → ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom ℎ ∩ dom 𝑔 ) ∧ ∀ 𝑎 ∈ ( dom ℎ ∩ dom 𝑔 ) ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) ) ) )
29 28 ancoms ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom ℎ ∩ dom 𝑔 ) ∧ ∀ 𝑎 ∈ ( dom ℎ ∩ dom 𝑔 ) ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) ) ) )
30 incom ⊢ ( dom 𝑔 ∩ dom ℎ ) = ( dom ℎ ∩ dom 𝑔 )
31 30 reseq2i ⊢ ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) = ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) )
32 31 fneq1i ⊢ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ↔ ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom 𝑔 ∩ dom ℎ ) )
33 30 fneq2i ⊢ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ↔ ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom ℎ ∩ dom 𝑔 ) )
34 32 33 bitri ⊢ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ↔ ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom ℎ ∩ dom 𝑔 ) )
35 31 fveq1i ⊢ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 )
36 predeq2 ⊢ ( ( dom 𝑔 ∩ dom ℎ ) = ( dom ℎ ∩ dom 𝑔 ) → Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) = Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) )
37 30 36 ax-mp ⊢ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) = Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 )
38 31 37 reseq12i ⊢ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) )
39 38 fveq2i ⊢ ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) )
40 35 39 eqeq12i ⊢ ( ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ↔ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) ) )
41 30 40 raleqbii ⊢ ( ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ↔ ∀ 𝑎 ∈ ( dom ℎ ∩ dom 𝑔 ) ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) ) )
42 34 41 anbi12i ⊢ ( ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) ↔ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) Fn ( dom ℎ ∩ dom 𝑔 ) ∧ ∀ 𝑎 ∈ ( dom ℎ ∩ dom 𝑔 ) ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom ℎ ∩ dom 𝑔 ) ) ↾ Pred ( 𝑅 , ( dom ℎ ∩ dom 𝑔 ) , 𝑎 ) ) ) ) )
43 29 42 sylibr ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) )
44 wfr3g ⊢ ( ( ( 𝑅 We ( dom 𝑔 ∩ dom ℎ ) ∧ 𝑅 Se ( dom 𝑔 ∩ dom ℎ ) ) ∧ ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) ∧ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝐹 ‘ ( ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
45 26 27 43 44 syl3anc ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
46 45 breqd ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ↔ 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) )
47 46 biimprd ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 → 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) )
48 3 wfrlem2 ⊢ ( 𝑔 ∈ 𝐵 → Fun 𝑔 )
49 funres ⊢ ( Fun 𝑔 → Fun ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
50 dffun2 ⊢ ( Fun ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↔ ( Rel ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ∧ ∀ 𝑥 ∀ 𝑢 ∀ 𝑣 ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) ) )
51 50 simprbi ⊢ ( Fun ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) → ∀ 𝑥 ∀ 𝑢 ∀ 𝑣 ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
52 2sp ⊢ ( ∀ 𝑢 ∀ 𝑣 ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) → ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
53 52 sps ⊢ ( ∀ 𝑥 ∀ 𝑢 ∀ 𝑣 ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) → ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
54 48 49 51 53 4syl ⊢ ( 𝑔 ∈ 𝐵 → ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
55 54 adantr ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
56 47 55 sylan2d ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑥 ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑢 ∧ 𝑥 ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) 𝑣 ) → 𝑢 = 𝑣 ) )
57 17 56 syl5 ⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )