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Theorem ctinfomlemom 12420
Description: Lemma for ctinfom 12421. Converting between ω and 0. (Contributed by Jim Kingdon, 10-Aug-2023.)
Hypotheses
Ref Expression
ctinfom.n 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ctinfom.g 𝐺 = (𝐹𝑁)
ctinfom.f (𝜑𝐹:ω–onto𝐴)
ctinfom.inf (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹𝑛))
Assertion
Ref Expression
ctinfomlemom (𝜑 → (𝐺:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖)))
Distinct variable groups:   𝑖,𝐹,𝑥   𝑛,𝐹   𝑗,𝐺,𝑘   𝑖,𝑁,𝑗,𝑘   𝑛,𝑁,𝑘   𝑥,𝑁,𝑘   𝑖,𝑚,𝑗,𝑘   𝜑,𝑖,𝑘,𝑚,𝑥   𝑚,𝑛
Allowed substitution hints:   𝜑(𝑗,𝑛)   𝐴(𝑥,𝑖,𝑗,𝑘,𝑚,𝑛)   𝐹(𝑗,𝑘,𝑚)   𝐺(𝑥,𝑖,𝑚,𝑛)   𝑁(𝑚)

Proof of Theorem ctinfomlemom
StepHypRef Expression
1 ctinfom.f . . . 4 (𝜑𝐹:ω–onto𝐴)
2 ctinfom.n . . . . . . 7 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
32frechashgf1o 10423 . . . . . 6 𝑁:ω–1-1-onto→ℕ0
4 f1ocnv 5473 . . . . . 6 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
53, 4ax-mp 5 . . . . 5 𝑁:ℕ01-1-onto→ω
6 f1ofo 5467 . . . . 5 (𝑁:ℕ01-1-onto→ω → 𝑁:ℕ0onto→ω)
75, 6ax-mp 5 . . . 4 𝑁:ℕ0onto→ω
8 foco 5447 . . . 4 ((𝐹:ω–onto𝐴𝑁:ℕ0onto→ω) → (𝐹𝑁):ℕ0onto𝐴)
91, 7, 8sylancl 413 . . 3 (𝜑 → (𝐹𝑁):ℕ0onto𝐴)
10 ctinfom.g . . . 4 𝐺 = (𝐹𝑁)
11 foeq1 5433 . . . 4 (𝐺 = (𝐹𝑁) → (𝐺:ℕ0onto𝐴 ↔ (𝐹𝑁):ℕ0onto𝐴))
1210, 11ax-mp 5 . . 3 (𝐺:ℕ0onto𝐴 ↔ (𝐹𝑁):ℕ0onto𝐴)
139, 12sylibr 134 . 2 (𝜑𝐺:ℕ0onto𝐴)
14 imaeq2 4965 . . . . . . . 8 (𝑛 = suc (𝑁𝑚) → (𝐹𝑛) = (𝐹 “ suc (𝑁𝑚)))
1514eleq2d 2247 . . . . . . 7 (𝑛 = suc (𝑁𝑚) → ((𝐹𝑘) ∈ (𝐹𝑛) ↔ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚))))
1615notbid 667 . . . . . 6 (𝑛 = suc (𝑁𝑚) → (¬ (𝐹𝑘) ∈ (𝐹𝑛) ↔ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚))))
1716rexbidv 2478 . . . . 5 (𝑛 = suc (𝑁𝑚) → (∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹𝑛) ↔ ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚))))
18 ctinfom.inf . . . . . 6 (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹𝑛))
1918adantr 276 . . . . 5 ((𝜑𝑚 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹𝑛))
20 f1of 5460 . . . . . . . . 9 (𝑁:ℕ01-1-onto→ω → 𝑁:ℕ0⟶ω)
215, 20ax-mp 5 . . . . . . . 8 𝑁:ℕ0⟶ω
2221a1i 9 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → 𝑁:ℕ0⟶ω)
23 simpr 110 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
2422, 23ffvelcdmd 5651 . . . . . 6 ((𝜑𝑚 ∈ ℕ0) → (𝑁𝑚) ∈ ω)
25 peano2 4593 . . . . . 6 ((𝑁𝑚) ∈ ω → suc (𝑁𝑚) ∈ ω)
2624, 25syl 14 . . . . 5 ((𝜑𝑚 ∈ ℕ0) → suc (𝑁𝑚) ∈ ω)
2717, 19, 26rspcdva 2846 . . . 4 ((𝜑𝑚 ∈ ℕ0) → ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))
28 f1of 5460 . . . . . . . 8 (𝑁:ω–1-1-onto→ℕ0𝑁:ω⟶ℕ0)
293, 28ax-mp 5 . . . . . . 7 𝑁:ω⟶ℕ0
3029a1i 9 . . . . . 6 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → 𝑁:ω⟶ℕ0)
31 simprl 529 . . . . . 6 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → 𝑘 ∈ ω)
3230, 31ffvelcdmd 5651 . . . . 5 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → (𝑁𝑘) ∈ ℕ0)
3310fveq1i 5515 . . . . . . . . . . 11 (𝐺‘(𝑁𝑘)) = ((𝐹𝑁)‘(𝑁𝑘))
3432adantr 276 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑘) ∈ ℕ0)
35 fvco3 5586 . . . . . . . . . . . 12 ((𝑁:ℕ0⟶ω ∧ (𝑁𝑘) ∈ ℕ0) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3621, 34, 35sylancr 414 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3733, 36eqtrid 2222 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3831adantr 276 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑘 ∈ ω)
39 f1ocnvfv1 5775 . . . . . . . . . . . . 13 ((𝑁:ω–1-1-onto→ℕ0𝑘 ∈ ω) → (𝑁‘(𝑁𝑘)) = 𝑘)
403, 39mpan 424 . . . . . . . . . . . 12 (𝑘 ∈ ω → (𝑁‘(𝑁𝑘)) = 𝑘)
4140fveq2d 5518 . . . . . . . . . . 11 (𝑘 ∈ ω → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
4238, 41syl 14 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐹‘(𝑁‘(𝑁𝑘))) = (𝐹𝑘))
4337, 42eqtrd 2210 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) = (𝐹𝑘))
44 simplrr 536 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))
4543, 44eqneltrd 2273 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁𝑘)) ∈ (𝐹 “ suc (𝑁𝑚)))
46 simpr 110 . . . . . . . . 9 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺‘(𝑁𝑘)) = (𝐺𝑖))
4710fveq1i 5515 . . . . . . . . . . . 12 (𝐺𝑖) = ((𝐹𝑁)‘𝑖)
48 elfznn0 10109 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑚) → 𝑖 ∈ ℕ0)
4948adantl 277 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ∈ ℕ0)
50 fvco3 5586 . . . . . . . . . . . . 13 ((𝑁:ℕ0⟶ω ∧ 𝑖 ∈ ℕ0) → ((𝐹𝑁)‘𝑖) = (𝐹‘(𝑁𝑖)))
5121, 49, 50sylancr 414 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹𝑁)‘𝑖) = (𝐹‘(𝑁𝑖)))
5247, 51eqtrid 2222 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺𝑖) = (𝐹‘(𝑁𝑖)))
53 elfzle2 10023 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0...𝑚) → 𝑖𝑚)
5453adantl 277 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖𝑚)
55 0zd 9261 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 0 ∈ ℤ)
5621a1i 9 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑁:ℕ0⟶ω)
5756, 49ffvelcdmd 5651 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ ω)
5824ad2antrr 488 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑚) ∈ ω)
5955, 2, 57, 58frec2uzled 10424 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁‘(𝑁𝑖)) ≤ (𝑁‘(𝑁𝑚))))
60 f1ocnvfv2 5776 . . . . . . . . . . . . . . . . 17 ((𝑁:ω–1-1-onto→ℕ0𝑖 ∈ ℕ0) → (𝑁‘(𝑁𝑖)) = 𝑖)
613, 49, 60sylancr 414 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(𝑁𝑖)) = 𝑖)
6223ad2antrr 488 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑚 ∈ ℕ0)
63 f1ocnvfv2 5776 . . . . . . . . . . . . . . . . 17 ((𝑁:ω–1-1-onto→ℕ0𝑚 ∈ ℕ0) → (𝑁‘(𝑁𝑚)) = 𝑚)
643, 62, 63sylancr 414 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(𝑁𝑚)) = 𝑚)
6561, 64breq12d 4015 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁‘(𝑁𝑖)) ≤ (𝑁‘(𝑁𝑚)) ↔ 𝑖𝑚))
6659, 65bitrd 188 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ 𝑖𝑚))
6754, 66mpbird 167 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ⊆ (𝑁𝑚))
68 nnsssuc 6500 . . . . . . . . . . . . . 14 (((𝑁𝑖) ∈ ω ∧ (𝑁𝑚) ∈ ω) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁𝑖) ∈ suc (𝑁𝑚)))
6957, 58, 68syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁𝑖) ∈ suc (𝑁𝑚)))
7067, 69mpbid 147 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ suc (𝑁𝑚))
711ad3antrrr 492 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω–onto𝐴)
72 fof 5437 . . . . . . . . . . . . . . 15 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
7371, 72syl 14 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω⟶𝐴)
7473ffund 5368 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
7573fdmd 5371 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
7657, 75eleqtrrd 2257 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ dom 𝐹)
77 funfvima 5746 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑁𝑖) ∈ dom 𝐹) → ((𝑁𝑖) ∈ suc (𝑁𝑚) → (𝐹‘(𝑁𝑖)) ∈ (𝐹 “ suc (𝑁𝑚))))
7874, 76, 77syl2anc 411 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ∈ suc (𝑁𝑚) → (𝐹‘(𝑁𝑖)) ∈ (𝐹 “ suc (𝑁𝑚))))
7970, 78mpd 13 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐹‘(𝑁𝑖)) ∈ (𝐹 “ suc (𝑁𝑚)))
8052, 79eqeltrd 2254 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺𝑖) ∈ (𝐹 “ suc (𝑁𝑚)))
8180adantr 276 . . . . . . . . 9 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺𝑖) ∈ (𝐹 “ suc (𝑁𝑚)))
8246, 81eqeltrd 2254 . . . . . . . 8 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺‘(𝑁𝑘)) ∈ (𝐹 “ suc (𝑁𝑚)))
8345, 82mtand 665 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁𝑘)) = (𝐺𝑖))
8483neqned 2354 . . . . . 6 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖))
8584ralrimiva 2550 . . . . 5 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖))
86 fveq2 5514 . . . . . . . 8 (𝑗 = (𝑁𝑘) → (𝐺𝑗) = (𝐺‘(𝑁𝑘)))
8786neeq1d 2365 . . . . . . 7 (𝑗 = (𝑁𝑘) → ((𝐺𝑗) ≠ (𝐺𝑖) ↔ (𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)))
8887ralbidv 2477 . . . . . 6 (𝑗 = (𝑁𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)))
8988rspcev 2841 . . . . 5 (((𝑁𝑘) ∈ ℕ0 ∧ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9032, 85, 89syl2anc 411 . . . 4 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9127, 90rexlimddv 2599 . . 3 ((𝜑𝑚 ∈ ℕ0) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9291ralrimiva 2550 . 2 (𝜑 → ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9313, 92jca 306 1 (𝜑 → (𝐺:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  wss 3129   class class class wbr 4002  cmpt 4063  suc csuc 4364  ωcom 4588  ccnv 4624  dom cdm 4625  cima 4628  ccom 4629  Fun wfun 5209  wf 5211  ontowfo 5213  1-1-ontowf1o 5214  cfv 5215  (class class class)co 5872  freccfrec 6388  0cc0 7808  1c1 7809   + caddc 7811  cle 7989  0cn0 9172  cz 9249  ...cfz 10004
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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  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-nel 2443  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-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-recs 6303  df-frec 6389  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-inn 8916  df-n0 9173  df-z 9250  df-uz 9525  df-fz 10005
This theorem is referenced by:  ctinfom  12421
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