Metamath Proof Explorer

Theorem mh-inf3f1

Description: A variant of inf3 . If F is a one-to-one function from A into itself, and B is an element outside its range, then ` ( rec ( F , B ) |`_om ) is a one-to-one function yielding an infinite sequence of distinct elements from A . If A is a set, we can use this theorem to prove _om e. _V via f1dmex . (Contributed by Matthew House, 13-Apr-2026)

Ref Expression
Hypotheses mh-inf3f1.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1→ 𝐴 )
mh-inf3f1.2 ⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 ∖ ran 𝐹 ) )
Assertion mh-inf3f1 ( 𝜑 → ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω –1-1→ 𝐴 )

Proof

Step Hyp Ref Expression
1 mh-inf3f1.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1→ 𝐴 )
2 mh-inf3f1.2 ⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 ∖ ran 𝐹 ) )
3 fveq2 ⊢ ( 𝑥 = ∅ → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) )
4 3 eleq1d ⊢ ( 𝑥 = ∅ → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ∈ 𝐴 ) )
5 fveq2 ⊢ ( 𝑥 = 𝑧 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) )
6 5 eleq1d ⊢ ( 𝑥 = 𝑧 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 ) )
7 fveq2 ⊢ ( 𝑥 = suc 𝑧 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) )
8 7 eleq1d ⊢ ( 𝑥 = suc 𝑧 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ∈ 𝐴 ) )
9 fr0g ⊢ ( 𝐵 ∈ ( 𝐴 ∖ ran 𝐹 ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) = 𝐵 )
10 2 9 syl ⊢ ( 𝜑 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) = 𝐵 )
11 10 2 eqeltrd ⊢ ( 𝜑 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ∈ ( 𝐴 ∖ ran 𝐹 ) )
12 11 eldifad ⊢ ( 𝜑 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ∈ 𝐴 )
13 f1f ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐴 → 𝐹 : 𝐴 ⟶ 𝐴 )
14 1 13 syl ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐴 )
15 14 ffvelcdmda ⊢ ( ( 𝜑 ∧ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ∈ 𝐴 )
16 frsuc ⊢ ( 𝑧 ∈ ω → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) )
17 16 eleq1d ⊢ ( 𝑧 ∈ ω → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ∈ 𝐴 ↔ ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ∈ 𝐴 ) )
18 15 17 imbitrrid ⊢ ( 𝑧 ∈ ω → ( ( 𝜑 ∧ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ∈ 𝐴 ) )
19 18 expd ⊢ ( 𝑧 ∈ ω → ( 𝜑 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ∈ 𝐴 ) ) )
20 4 6 8 12 19 finds2 ⊢ ( 𝑥 ∈ ω → ( 𝜑 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ) )
21 20 com12 ⊢ ( 𝜑 → ( 𝑥 ∈ ω → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ) )
22 21 ralrimiv ⊢ ( 𝜑 → ∀ 𝑥 ∈ ω ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 )
23 frfnom ⊢ ( rec ( 𝐹 , 𝐵 ) ↾ ω ) Fn ω
24 ffnfv ⊢ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω ⟶ 𝐴 ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) Fn ω ∧ ∀ 𝑥 ∈ ω ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ) )
25 23 24 mpbiran ⊢ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω ⟶ 𝐴 ↔ ∀ 𝑥 ∈ ω ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 )
26 22 25 sylibr ⊢ ( 𝜑 → ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω ⟶ 𝐴 )
27 3 neeq1d ⊢ ( 𝑥 = ∅ → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
28 27 raleqbi1dv ⊢ ( 𝑥 = ∅ → ( ∀ 𝑦 ∈ 𝑥 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ∅ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
29 5 neeq1d ⊢ ( 𝑥 = 𝑧 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
30 29 raleqbi1dv ⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝑥 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
31 7 neeq1d ⊢ ( 𝑥 = suc 𝑧 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
32 31 raleqbi1dv ⊢ ( 𝑥 = suc 𝑧 → ( ∀ 𝑦 ∈ 𝑥 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ suc 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
33 ral0 ⊢ ∀ 𝑦 ∈ ∅ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 )
34 33 a1i ⊢ ( 𝜑 → ∀ 𝑦 ∈ ∅ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
35 nfv ⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑧 ∈ ω )
36 nfra1 ⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 )
37 35 36 nfan ⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
38 16 ad3antlr ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) )
39 fveq2 ⊢ ( 𝑦 = ∅ → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) )
40 39 neeq2d ⊢ ( 𝑦 = ∅ → ( ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ) )
41 peano2b ⊢ ( 𝑧 ∈ ω ↔ suc 𝑧 ∈ ω )
42 elnn ⊢ ( ( 𝑦 ∈ suc 𝑧 ∧ suc 𝑧 ∈ ω ) → 𝑦 ∈ ω )
43 42 ancoms ⊢ ( ( suc 𝑧 ∈ ω ∧ 𝑦 ∈ suc 𝑧 ) → 𝑦 ∈ ω )
44 41 43 sylanb ⊢ ( ( 𝑧 ∈ ω ∧ 𝑦 ∈ suc 𝑧 ) → 𝑦 ∈ ω )
45 44 ad4ant24 ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) → 𝑦 ∈ ω )
46 nnsuc ⊢ ( ( 𝑦 ∈ ω ∧ 𝑦 ≠ ∅ ) → ∃ 𝑥 ∈ ω 𝑦 = suc 𝑥 )
47 45 46 sylan ⊢ ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) → ∃ 𝑥 ∈ ω 𝑦 = suc 𝑥 )
48 fveq2 ⊢ ( 𝑦 = 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) )
49 48 neeq2d ⊢ ( 𝑦 = 𝑥 → ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
50 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
51 simprr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝑦 = suc 𝑥 )
52 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝑦 ∈ suc 𝑧 )
53 51 52 eqeltrrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → suc 𝑥 ∈ suc 𝑧 )
54 nnord ⊢ ( 𝑧 ∈ ω → Ord 𝑧 )
55 54 ad5antlr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → Ord 𝑧 )
56 ordsucelsuc ⊢ ( Ord 𝑧 → ( 𝑥 ∈ 𝑧 ↔ suc 𝑥 ∈ suc 𝑧 ) )
57 55 56 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( 𝑥 ∈ 𝑧 ↔ suc 𝑥 ∈ suc 𝑧 ) )
58 53 57 mpbird ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝑥 ∈ 𝑧 )
59 49 50 58 rspcdva ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) )
60 simp-5l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝜑 )
61 60 1 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝐹 : 𝐴 –1-1→ 𝐴 )
62 26 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ω ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 )
63 62 ad4antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 )
64 simprl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → 𝑥 ∈ ω )
65 64 60 20 sylc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 )
66 f1fveq ⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 ∧ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
67 66 necon3bid ⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ∈ 𝐴 ∧ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
68 61 63 65 67 syl12anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) ↔ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
69 59 68 mpbird ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
70 fveq2 ⊢ ( 𝑦 = suc 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) = ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑥 ) )
71 frsuc ⊢ ( 𝑥 ∈ ω → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑥 ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
72 70 71 sylan9eqr ⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
73 72 adantl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ) )
74 69 73 neeqtrrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 = suc 𝑥 ) ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
75 47 74 rexlimddv ⊢ ( ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) ∧ 𝑦 ≠ ∅ ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
76 14 ffnd ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
77 76 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ω ) → 𝐹 Fn 𝐴 )
78 77 62 fnfvelrnd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ω ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ∈ ran 𝐹 )
79 11 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ω ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ∈ ( 𝐴 ∖ ran 𝐹 ) )
80 elneeldif ⊢ ( ( ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ∈ ran 𝐹 ∧ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) ∈ ( 𝐴 ∖ ran 𝐹 ) ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) )
81 78 79 80 syl2anc ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ω ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) )
82 81 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ ∅ ) )
83 40 75 82 pm2.61ne ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) → ( 𝐹 ‘ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
84 38 83 eqnetrd ⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ∧ 𝑦 ∈ suc 𝑧 ) → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
85 37 84 ralrimia ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ω ) ∧ ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) → ∀ 𝑦 ∈ suc 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) )
86 85 exp31 ⊢ ( 𝜑 → ( 𝑧 ∈ ω → ( ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) → ∀ 𝑦 ∈ suc 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) )
87 86 com12 ⊢ ( 𝑧 ∈ ω → ( 𝜑 → ( ∀ 𝑦 ∈ 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) → ∀ 𝑦 ∈ suc 𝑧 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ suc 𝑧 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) )
88 28 30 32 34 87 finds2 ⊢ ( 𝑥 ∈ ω → ( 𝜑 → ∀ 𝑦 ∈ 𝑥 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
89 rsp ⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) → ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
90 88 89 syl6com ⊢ ( 𝜑 → ( 𝑥 ∈ ω → ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) )
91 90 adantrd ⊢ ( 𝜑 → ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) )
92 91 ralrimivv ⊢ ( 𝜑 → ∀ 𝑥 ∈ ω ∀ 𝑦 ∈ ω ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) )
93 omsson ⊢ ω ⊆ On
94 onelfvnef1 ⊢ ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω ⟶ 𝐴 ∧ ω ⊆ On ∧ ∀ 𝑥 ∈ ω ∀ 𝑦 ∈ ω ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) → ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω –1-1→ 𝐴 )
95 93 94 mp3an2 ⊢ ( ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω ⟶ 𝐴 ∧ ∀ 𝑥 ∈ ω ∀ 𝑦 ∈ ω ( 𝑦 ∈ 𝑥 → ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑥 ) ≠ ( ( rec ( 𝐹 , 𝐵 ) ↾ ω ) ‘ 𝑦 ) ) ) → ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω –1-1→ 𝐴 )
96 26 92 95 syl2anc ⊢ ( 𝜑 → ( rec ( 𝐹 , 𝐵 ) ↾ ω ) : ω –1-1→ 𝐴 )