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Theorem peano4nninf 14378
Description: The successor function on is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
Hypothesis
Ref Expression
peano4nninf.s 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
Assertion
Ref Expression
peano4nninf 𝑆:ℕ1-1→ℕ
Distinct variable groups:   𝑆,𝑖   𝑖,𝑝
Allowed substitution hint:   𝑆(𝑝)

Proof of Theorem peano4nninf
Dummy variables 𝑘 𝑥 𝑦 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano4nninf.s . . 3 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
21nnsf 14377 . 2 𝑆:ℕ⟶ℕ
3 fveq1 5509 . . . . . . . . . . 11 (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4 fveq1 5509 . . . . . . . . . . 11 (𝑓 = 𝑥 → (𝑓𝑗) = (𝑥𝑗))
53, 4sseq12d 3186 . . . . . . . . . 10 (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
65ralbidv 2477 . . . . . . . . 9 (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
7 df-nninf 7112 . . . . . . . . 9 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
86, 7elrab2 2896 . . . . . . . 8 (𝑥 ∈ ℕ ↔ (𝑥 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
98simplbi 274 . . . . . . 7 (𝑥 ∈ ℕ𝑥 ∈ (2o𝑚 ω))
10 elmapfn 6664 . . . . . . 7 (𝑥 ∈ (2o𝑚 ω) → 𝑥 Fn ω)
119, 10syl 14 . . . . . 6 (𝑥 ∈ ℕ𝑥 Fn ω)
1211ad2antrr 488 . . . . 5 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑥 Fn ω)
13 fveq1 5509 . . . . . . . . . . 11 (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14 fveq1 5509 . . . . . . . . . . 11 (𝑓 = 𝑦 → (𝑓𝑗) = (𝑦𝑗))
1513, 14sseq12d 3186 . . . . . . . . . 10 (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1615ralbidv 2477 . . . . . . . . 9 (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1716, 7elrab2 2896 . . . . . . . 8 (𝑦 ∈ ℕ ↔ (𝑦 ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1817simplbi 274 . . . . . . 7 (𝑦 ∈ ℕ𝑦 ∈ (2o𝑚 ω))
19 elmapfn 6664 . . . . . . 7 (𝑦 ∈ (2o𝑚 ω) → 𝑦 Fn ω)
2018, 19syl 14 . . . . . 6 (𝑦 ∈ ℕ𝑦 Fn ω)
2120ad2antlr 489 . . . . 5 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑦 Fn ω)
22 simplr 528 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑥) = (𝑆𝑦))
2322fveq1d 5512 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = ((𝑆𝑦)‘suc 𝑘))
24 fveq1 5509 . . . . . . . . . . . 12 (𝑝 = 𝑥 → (𝑝 𝑖) = (𝑥 𝑖))
2524ifeq2d 3552 . . . . . . . . . . 11 (𝑝 = 𝑥 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑖 = ∅, 1o, (𝑥 𝑖)))
2625mpteq2dv 4091 . . . . . . . . . 10 (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥 𝑖))))
27 omex 4588 . . . . . . . . . . 11 ω ∈ V
2827mptex 5737 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥 𝑖))) ∈ V
2926, 1, 28fvmpt 5588 . . . . . . . . 9 (𝑥 ∈ ℕ → (𝑆𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥 𝑖))))
3029ad3antrrr 492 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥 𝑖))))
31 simpr 110 . . . . . . . . . 10 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
3231eqeq1d 2186 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
3331unieqd 3818 . . . . . . . . . 10 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
3433fveq2d 5514 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥 𝑖) = (𝑥 suc 𝑘))
3532, 34ifbieq2d 3558 . . . . . . . 8 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → if(𝑖 = ∅, 1o, (𝑥 𝑖)) = if(suc 𝑘 = ∅, 1o, (𝑥 suc 𝑘)))
36 peano2 4590 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
3736adantl 277 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
38 1lt2o 6436 . . . . . . . . . 10 1o ∈ 2o
3938a1i 9 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40 nninff 7114 . . . . . . . . . . 11 (𝑥 ∈ ℕ𝑥:ω⟶2o)
4140ad3antrrr 492 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2o)
42 nnpredcl 4618 . . . . . . . . . . 11 (suc 𝑘 ∈ ω → suc 𝑘 ∈ ω)
4337, 42syl 14 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
4441, 43ffvelcdmd 5647 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥 suc 𝑘) ∈ 2o)
45 nndceq0 4613 . . . . . . . . . 10 (suc 𝑘 ∈ ω → DECID suc 𝑘 = ∅)
4637, 45syl 14 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → DECID suc 𝑘 = ∅)
4739, 44, 46ifcldcd 3569 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥 suc 𝑘)) ∈ 2o)
4830, 35, 37, 47fvmptd 5592 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥 suc 𝑘)))
49 peano3 4591 . . . . . . . . . 10 (𝑘 ∈ ω → suc 𝑘 ≠ ∅)
5049adantl 277 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ≠ ∅)
5150neneqd 2368 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
5251iffalsed 3544 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑥 suc 𝑘)) = (𝑥 suc 𝑘))
53 nnord 4607 . . . . . . . . . . 11 (𝑘 ∈ ω → Ord 𝑘)
54 ordtr 4374 . . . . . . . . . . 11 (Ord 𝑘 → Tr 𝑘)
5553, 54syl 14 . . . . . . . . . 10 (𝑘 ∈ ω → Tr 𝑘)
56 unisucg 4410 . . . . . . . . . 10 (𝑘 ∈ ω → (Tr 𝑘 suc 𝑘 = 𝑘))
5755, 56mpbid 147 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 = 𝑘)
5857fveq2d 5514 . . . . . . . 8 (𝑘 ∈ ω → (𝑥 suc 𝑘) = (𝑥𝑘))
5958adantl 277 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥 suc 𝑘) = (𝑥𝑘))
6048, 52, 593eqtrd 2214 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = (𝑥𝑘))
61 fveq1 5509 . . . . . . . . . . . 12 (𝑝 = 𝑦 → (𝑝 𝑖) = (𝑦 𝑖))
6261ifeq2d 3552 . . . . . . . . . . 11 (𝑝 = 𝑦 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑖 = ∅, 1o, (𝑦 𝑖)))
6362mpteq2dv 4091 . . . . . . . . . 10 (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦 𝑖))))
6427mptex 5737 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦 𝑖))) ∈ V
6563, 1, 64fvmpt 5588 . . . . . . . . 9 (𝑦 ∈ ℕ → (𝑆𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦 𝑖))))
6665ad3antlr 493 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦 𝑖))))
6733fveq2d 5514 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦 𝑖) = (𝑦 suc 𝑘))
6832, 67ifbieq2d 3558 . . . . . . . 8 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → if(𝑖 = ∅, 1o, (𝑦 𝑖)) = if(suc 𝑘 = ∅, 1o, (𝑦 suc 𝑘)))
69 nninff 7114 . . . . . . . . . . 11 (𝑦 ∈ ℕ𝑦:ω⟶2o)
7069ad3antlr 493 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2o)
7170, 43ffvelcdmd 5647 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑦 suc 𝑘) ∈ 2o)
7239, 71, 46ifcldcd 3569 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦 suc 𝑘)) ∈ 2o)
7366, 68, 37, 72fvmptd 5592 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑦)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑦 suc 𝑘)))
7451iffalsed 3544 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1o, (𝑦 suc 𝑘)) = (𝑦 suc 𝑘))
7557fveq2d 5514 . . . . . . . 8 (𝑘 ∈ ω → (𝑦 suc 𝑘) = (𝑦𝑘))
7675adantl 277 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑦 suc 𝑘) = (𝑦𝑘))
7773, 74, 763eqtrd 2214 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑦)‘suc 𝑘) = (𝑦𝑘))
7823, 60, 773eqtr3d 2218 . . . . 5 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥𝑘) = (𝑦𝑘))
7912, 21, 78eqfnfvd 5611 . . . 4 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑥 = 𝑦)
8079ex 115 . . 3 ((𝑥 ∈ ℕ𝑦 ∈ ℕ) → ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦))
8180rgen2a 2531 . 2 𝑥 ∈ ℕ𝑦 ∈ ℕ ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦)
82 dff13 5762 . 2 (𝑆:ℕ1-1→ℕ ↔ (𝑆:ℕ⟶ℕ ∧ ∀𝑥 ∈ ℕ𝑦 ∈ ℕ ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦)))
832, 81, 82mpbir2an 942 1 𝑆:ℕ1-1→ℕ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wss 3129  c0 3422  ifcif 3534   cuni 3807  cmpt 4061  Tr wtr 4098  Ord word 4358  suc csuc 4361  ωcom 4585   Fn wfn 5206  wf 5207  1-1wf1 5208  cfv 5211  (class class class)co 5868  1oc1o 6403  2oc2o 6404  𝑚 cmap 6641  xnninf 7111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1o 6410  df-2o 6411  df-map 6643  df-nninf 7112
This theorem is referenced by:  exmidsbthrlem  14393
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