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Theorem peano4nninf 14411
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 14410 . 2 𝑆:ℕ⟶ℕ
3 fveq1 5510 . . . . . . . . . . 11 (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4 fveq1 5510 . . . . . . . . . . 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 7113 . . . . . . . . 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 6665 . . . . . . 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 5510 . . . . . . . . . . 11 (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14 fveq1 5510 . . . . . . . . . . 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 6665 . . . . . . 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 5513 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = ((𝑆𝑦)‘suc 𝑘))
24 fveq1 5510 . . . . . . . . . . . 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 4589 . . . . . . . . . . 11 ω ∈ V
2827mptex 5738 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑥 𝑖))) ∈ V
2926, 1, 28fvmpt 5589 . . . . . . . . 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 5515 . . . . . . . . 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 4591 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
3736adantl 277 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
38 1lt2o 6437 . . . . . . . . . 10 1o ∈ 2o
3938a1i 9 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 1o ∈ 2o)
40 nninff 7115 . . . . . . . . . . 11 (𝑥 ∈ ℕ𝑥:ω⟶2o)
4140ad3antrrr 492 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2o)
42 nnpredcl 4619 . . . . . . . . . . 11 (suc 𝑘 ∈ ω → suc 𝑘 ∈ ω)
4337, 42syl 14 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
4441, 43ffvelcdmd 5648 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥 suc 𝑘) ∈ 2o)
45 nndceq0 4614 . . . . . . . . . 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 5593 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1o, (𝑥 suc 𝑘)))
49 peano3 4592 . . . . . . . . . 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 4608 . . . . . . . . . . 11 (𝑘 ∈ ω → Ord 𝑘)
54 ordtr 4375 . . . . . . . . . . 11 (Ord 𝑘 → Tr 𝑘)
5553, 54syl 14 . . . . . . . . . 10 (𝑘 ∈ ω → Tr 𝑘)
56 unisucg 4411 . . . . . . . . . 10 (𝑘 ∈ ω → (Tr 𝑘 suc 𝑘 = 𝑘))
5755, 56mpbid 147 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 = 𝑘)
5857fveq2d 5515 . . . . . . . 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 5510 . . . . . . . . . . . 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 5738 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑦 𝑖))) ∈ V
6563, 1, 64fvmpt 5589 . . . . . . . . 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 5515 . . . . . . . . 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 7115 . . . . . . . . . . 11 (𝑦 ∈ ℕ𝑦:ω⟶2o)
7069ad3antlr 493 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2o)
7170, 43ffvelcdmd 5648 . . . . . . . . 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 5593 . . . . . . 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 5515 . . . . . . . 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 5612 . . . 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 5763 . 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 4359  suc csuc 4362  ωcom 4586   Fn wfn 5207  wf 5208  1-1wf1 5209  cfv 5212  (class class class)co 5869  1oc1o 6404  2oc2o 6405  𝑚 cmap 6642  xnninf 7112
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 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1o 6411  df-2o 6412  df-map 6644  df-nninf 7113
This theorem is referenced by:  exmidsbthrlem  14426
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