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Theorem peano4nninf 11553
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 11552 . 2 𝑆:ℕ⟶ℕ
3 fveq1 5288 . . . . . . . . . . 11 (𝑓 = 𝑥 → (𝑓‘suc 𝑗) = (𝑥‘suc 𝑗))
4 fveq1 5288 . . . . . . . . . . 11 (𝑓 = 𝑥 → (𝑓𝑗) = (𝑥𝑗))
53, 4sseq12d 3053 . . . . . . . . . 10 (𝑓 = 𝑥 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
65ralbidv 2380 . . . . . . . . 9 (𝑓 = 𝑥 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
7 df-nninf 6770 . . . . . . . . 9 = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
86, 7elrab2 2772 . . . . . . . 8 (𝑥 ∈ ℕ ↔ (𝑥 ∈ (2𝑜𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑥‘suc 𝑗) ⊆ (𝑥𝑗)))
98simplbi 268 . . . . . . 7 (𝑥 ∈ ℕ𝑥 ∈ (2𝑜𝑚 ω))
10 elmapfn 6408 . . . . . . 7 (𝑥 ∈ (2𝑜𝑚 ω) → 𝑥 Fn ω)
119, 10syl 14 . . . . . 6 (𝑥 ∈ ℕ𝑥 Fn ω)
1211ad2antrr 472 . . . . 5 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑥 Fn ω)
13 fveq1 5288 . . . . . . . . . . 11 (𝑓 = 𝑦 → (𝑓‘suc 𝑗) = (𝑦‘suc 𝑗))
14 fveq1 5288 . . . . . . . . . . 11 (𝑓 = 𝑦 → (𝑓𝑗) = (𝑦𝑗))
1513, 14sseq12d 3053 . . . . . . . . . 10 (𝑓 = 𝑦 → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1615ralbidv 2380 . . . . . . . . 9 (𝑓 = 𝑦 → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1716, 7elrab2 2772 . . . . . . . 8 (𝑦 ∈ ℕ ↔ (𝑦 ∈ (2𝑜𝑚 ω) ∧ ∀𝑗 ∈ ω (𝑦‘suc 𝑗) ⊆ (𝑦𝑗)))
1817simplbi 268 . . . . . . 7 (𝑦 ∈ ℕ𝑦 ∈ (2𝑜𝑚 ω))
19 elmapfn 6408 . . . . . . 7 (𝑦 ∈ (2𝑜𝑚 ω) → 𝑦 Fn ω)
2018, 19syl 14 . . . . . 6 (𝑦 ∈ ℕ𝑦 Fn ω)
2120ad2antlr 473 . . . . 5 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑦 Fn ω)
22 simplr 497 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑥) = (𝑆𝑦))
2322fveq1d 5291 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = ((𝑆𝑦)‘suc 𝑘))
24 fveq1 5288 . . . . . . . . . . . 12 (𝑝 = 𝑥 → (𝑝 𝑖) = (𝑥 𝑖))
2524ifeq2d 3405 . . . . . . . . . . 11 (𝑝 = 𝑥 → if(𝑖 = ∅, 1𝑜, (𝑝 𝑖)) = if(𝑖 = ∅, 1𝑜, (𝑥 𝑖)))
2625mpteq2dv 3921 . . . . . . . . . 10 (𝑝 = 𝑥 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑥 𝑖))))
27 omex 4398 . . . . . . . . . . 11 ω ∈ V
2827mptex 5505 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑥 𝑖))) ∈ V
2926, 1, 28fvmpt 5365 . . . . . . . . 9 (𝑥 ∈ ℕ → (𝑆𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑥 𝑖))))
3029ad3antrrr 476 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑥) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑥 𝑖))))
31 simpr 108 . . . . . . . . . 10 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
3231eqeq1d 2096 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑖 = ∅ ↔ suc 𝑘 = ∅))
3331unieqd 3659 . . . . . . . . . 10 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → 𝑖 = suc 𝑘)
3433fveq2d 5293 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑥 𝑖) = (𝑥 suc 𝑘))
3532, 34ifbieq2d 3411 . . . . . . . 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 4400 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
3736adantl 271 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
38 1lt2o 11543 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
3938a1i 9 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 1𝑜 ∈ 2𝑜)
40 nninff 11551 . . . . . . . . . . 11 (𝑥 ∈ ℕ𝑥:ω⟶2𝑜)
4140ad3antrrr 476 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑥:ω⟶2𝑜)
42 nnpredcl 11547 . . . . . . . . . . 11 (suc 𝑘 ∈ ω → suc 𝑘 ∈ ω)
4337, 42syl 14 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ ω)
4441, 43ffvelrnd 5419 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥 suc 𝑘) ∈ 2𝑜)
45 nndceq0 4421 . . . . . . . . . 10 (suc 𝑘 ∈ ω → DECID suc 𝑘 = ∅)
4637, 45syl 14 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → DECID suc 𝑘 = ∅)
4739, 44, 46ifcldcd 3422 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1𝑜, (𝑥 suc 𝑘)) ∈ 2𝑜)
4830, 35, 37, 47fvmptd 5369 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = if(suc 𝑘 = ∅, 1𝑜, (𝑥 suc 𝑘)))
49 peano3 4401 . . . . . . . . . 10 (𝑘 ∈ ω → suc 𝑘 ≠ ∅)
5049adantl 271 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → suc 𝑘 ≠ ∅)
5150neneqd 2276 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ¬ suc 𝑘 = ∅)
5251iffalsed 3399 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1𝑜, (𝑥 suc 𝑘)) = (𝑥 suc 𝑘))
53 nnord 4416 . . . . . . . . . . 11 (𝑘 ∈ ω → Ord 𝑘)
54 ordtr 4196 . . . . . . . . . . 11 (Ord 𝑘 → Tr 𝑘)
5553, 54syl 14 . . . . . . . . . 10 (𝑘 ∈ ω → Tr 𝑘)
56 unisucg 4232 . . . . . . . . . 10 (𝑘 ∈ ω → (Tr 𝑘 suc 𝑘 = 𝑘))
5755, 56mpbid 145 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 = 𝑘)
5857fveq2d 5293 . . . . . . . 8 (𝑘 ∈ ω → (𝑥 suc 𝑘) = (𝑥𝑘))
5958adantl 271 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥 suc 𝑘) = (𝑥𝑘))
6048, 52, 593eqtrd 2124 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑥)‘suc 𝑘) = (𝑥𝑘))
61 fveq1 5288 . . . . . . . . . . . 12 (𝑝 = 𝑦 → (𝑝 𝑖) = (𝑦 𝑖))
6261ifeq2d 3405 . . . . . . . . . . 11 (𝑝 = 𝑦 → if(𝑖 = ∅, 1𝑜, (𝑝 𝑖)) = if(𝑖 = ∅, 1𝑜, (𝑦 𝑖)))
6362mpteq2dv 3921 . . . . . . . . . 10 (𝑝 = 𝑦 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑝 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑦 𝑖))))
6427mptex 5505 . . . . . . . . . 10 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑦 𝑖))) ∈ V
6563, 1, 64fvmpt 5365 . . . . . . . . 9 (𝑦 ∈ ℕ → (𝑆𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑦 𝑖))))
6665ad3antlr 477 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑆𝑦) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝑦 𝑖))))
6733fveq2d 5293 . . . . . . . . 9 (((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) ∧ 𝑖 = suc 𝑘) → (𝑦 𝑖) = (𝑦 suc 𝑘))
6832, 67ifbieq2d 3411 . . . . . . . 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 11551 . . . . . . . . . . 11 (𝑦 ∈ ℕ𝑦:ω⟶2𝑜)
7069ad3antlr 477 . . . . . . . . . 10 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → 𝑦:ω⟶2𝑜)
7170, 43ffvelrnd 5419 . . . . . . . . 9 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑦 suc 𝑘) ∈ 2𝑜)
7239, 71, 46ifcldcd 3422 . . . . . . . 8 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1𝑜, (𝑦 suc 𝑘)) ∈ 2𝑜)
7366, 68, 37, 72fvmptd 5369 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑦)‘suc 𝑘) = if(suc 𝑘 = ∅, 1𝑜, (𝑦 suc 𝑘)))
7451iffalsed 3399 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → if(suc 𝑘 = ∅, 1𝑜, (𝑦 suc 𝑘)) = (𝑦 suc 𝑘))
7557fveq2d 5293 . . . . . . . 8 (𝑘 ∈ ω → (𝑦 suc 𝑘) = (𝑦𝑘))
7675adantl 271 . . . . . . 7 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑦 suc 𝑘) = (𝑦𝑘))
7773, 74, 763eqtrd 2124 . . . . . 6 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → ((𝑆𝑦)‘suc 𝑘) = (𝑦𝑘))
7823, 60, 773eqtr3d 2128 . . . . 5 ((((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) ∧ 𝑘 ∈ ω) → (𝑥𝑘) = (𝑦𝑘))
7912, 21, 78eqfnfvd 5384 . . . 4 (((𝑥 ∈ ℕ𝑦 ∈ ℕ) ∧ (𝑆𝑥) = (𝑆𝑦)) → 𝑥 = 𝑦)
8079ex 113 . . 3 ((𝑥 ∈ ℕ𝑦 ∈ ℕ) → ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦))
8180rgen2a 2429 . 2 𝑥 ∈ ℕ𝑦 ∈ ℕ ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦)
82 dff13 5529 . 2 (𝑆:ℕ1-1→ℕ ↔ (𝑆:ℕ⟶ℕ ∧ ∀𝑥 ∈ ℕ𝑦 ∈ ℕ ((𝑆𝑥) = (𝑆𝑦) → 𝑥 = 𝑦)))
832, 81, 82mpbir2an 888 1 𝑆:ℕ1-1→ℕ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  DECID wdc 780   = wceq 1289  wcel 1438  wne 2255  wral 2359  wss 2997  c0 3284  ifcif 3389   cuni 3648  cmpt 3891  Tr wtr 3928  Ord word 4180  suc csuc 4183  ωcom 4395   Fn wfn 4997  wf 4998  1-1wf1 4999  cfv 5002  (class class class)co 5634  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  xnninf 6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770
This theorem is referenced by:  exmidsbthrlem  11569
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