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Theorem ctinfomlemom 11963
 Description: Lemma for ctinfom 11964. Converting between ω and ℕ0. (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 10225 . . . . . 6 𝑁:ω–1-1-onto→ℕ0
4 f1ocnv 5383 . . . . . 6 (𝑁:ω–1-1-onto→ℕ0𝑁:ℕ01-1-onto→ω)
53, 4ax-mp 5 . . . . 5 𝑁:ℕ01-1-onto→ω
6 f1ofo 5377 . . . . 5 (𝑁:ℕ01-1-onto→ω → 𝑁:ℕ0onto→ω)
75, 6ax-mp 5 . . . 4 𝑁:ℕ0onto→ω
8 foco 5358 . . . 4 ((𝐹:ω–onto𝐴𝑁:ℕ0onto→ω) → (𝐹𝑁):ℕ0onto𝐴)
91, 7, 8sylancl 409 . . 3 (𝜑 → (𝐹𝑁):ℕ0onto𝐴)
10 ctinfom.g . . . 4 𝐺 = (𝐹𝑁)
11 foeq1 5344 . . . 4 (𝐺 = (𝐹𝑁) → (𝐺:ℕ0onto𝐴 ↔ (𝐹𝑁):ℕ0onto𝐴))
1210, 11ax-mp 5 . . 3 (𝐺:ℕ0onto𝐴 ↔ (𝐹𝑁):ℕ0onto𝐴)
139, 12sylibr 133 . 2 (𝜑𝐺:ℕ0onto𝐴)
14 imaeq2 4880 . . . . . . . 8 (𝑛 = suc (𝑁𝑚) → (𝐹𝑛) = (𝐹 “ suc (𝑁𝑚)))
1514eleq2d 2209 . . . . . . 7 (𝑛 = suc (𝑁𝑚) → ((𝐹𝑘) ∈ (𝐹𝑛) ↔ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚))))
1615notbid 656 . . . . . 6 (𝑛 = suc (𝑁𝑚) → (¬ (𝐹𝑘) ∈ (𝐹𝑛) ↔ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚))))
1716rexbidv 2438 . . . . 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 274 . . . . 5 ((𝜑𝑚 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹𝑛))
20 f1of 5370 . . . . . . . . 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 109 . . . . . . 7 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
2422, 23ffvelrnd 5559 . . . . . 6 ((𝜑𝑚 ∈ ℕ0) → (𝑁𝑚) ∈ ω)
25 peano2 4512 . . . . . 6 ((𝑁𝑚) ∈ ω → suc (𝑁𝑚) ∈ ω)
2624, 25syl 14 . . . . 5 ((𝜑𝑚 ∈ ℕ0) → suc (𝑁𝑚) ∈ ω)
2717, 19, 26rspcdva 2794 . . . 4 ((𝜑𝑚 ∈ ℕ0) → ∃𝑘 ∈ ω ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))
28 f1of 5370 . . . . . . . 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 520 . . . . . 6 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → 𝑘 ∈ ω)
3230, 31ffvelrnd 5559 . . . . 5 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → (𝑁𝑘) ∈ ℕ0)
3310fveq1i 5425 . . . . . . . . . . 11 (𝐺‘(𝑁𝑘)) = ((𝐹𝑁)‘(𝑁𝑘))
3432adantr 274 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑘) ∈ ℕ0)
35 fvco3 5495 . . . . . . . . . . . 12 ((𝑁:ℕ0⟶ω ∧ (𝑁𝑘) ∈ ℕ0) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3621, 34, 35sylancr 410 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹𝑁)‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3733, 36syl5eq 2184 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) = (𝐹‘(𝑁‘(𝑁𝑘))))
3831adantr 274 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑘 ∈ ω)
39 f1ocnvfv1 5681 . . . . . . . . . . . . 13 ((𝑁:ω–1-1-onto→ℕ0𝑘 ∈ ω) → (𝑁‘(𝑁𝑘)) = 𝑘)
403, 39mpan 420 . . . . . . . . . . . 12 (𝑘 ∈ ω → (𝑁‘(𝑁𝑘)) = 𝑘)
4140fveq2d 5428 . . . . . . . . . . 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 2172 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) = (𝐹𝑘))
44 simplrr 525 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))
4543, 44eqneltrd 2235 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁𝑘)) ∈ (𝐹 “ suc (𝑁𝑚)))
46 simpr 109 . . . . . . . . 9 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺‘(𝑁𝑘)) = (𝐺𝑖))
4710fveq1i 5425 . . . . . . . . . . . 12 (𝐺𝑖) = ((𝐹𝑁)‘𝑖)
48 elfznn0 9918 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑚) → 𝑖 ∈ ℕ0)
4948adantl 275 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖 ∈ ℕ0)
50 fvco3 5495 . . . . . . . . . . . . 13 ((𝑁:ℕ0⟶ω ∧ 𝑖 ∈ ℕ0) → ((𝐹𝑁)‘𝑖) = (𝐹‘(𝑁𝑖)))
5121, 49, 50sylancr 410 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝐹𝑁)‘𝑖) = (𝐹‘(𝑁𝑖)))
5247, 51syl5eq 2184 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺𝑖) = (𝐹‘(𝑁𝑖)))
53 elfzle2 9832 . . . . . . . . . . . . . . 15 (𝑖 ∈ (0...𝑚) → 𝑖𝑚)
5453adantl 275 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑖𝑚)
55 0zd 9085 . . . . . . . . . . . . . . . 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, 49ffvelrnd 5559 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ ω)
5824ad2antrr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑚) ∈ ω)
5955, 2, 57, 58frec2uzled 10226 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁‘(𝑁𝑖)) ≤ (𝑁‘(𝑁𝑚))))
60 f1ocnvfv2 5682 . . . . . . . . . . . . . . . . 17 ((𝑁:ω–1-1-onto→ℕ0𝑖 ∈ ℕ0) → (𝑁‘(𝑁𝑖)) = 𝑖)
613, 49, 60sylancr 410 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(𝑁𝑖)) = 𝑖)
6223ad2antrr 479 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝑚 ∈ ℕ0)
63 f1ocnvfv2 5682 . . . . . . . . . . . . . . . . 17 ((𝑁:ω–1-1-onto→ℕ0𝑚 ∈ ℕ0) → (𝑁‘(𝑁𝑚)) = 𝑚)
643, 62, 63sylancr 410 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁‘(𝑁𝑚)) = 𝑚)
6561, 64breq12d 3945 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁‘(𝑁𝑖)) ≤ (𝑁‘(𝑁𝑚)) ↔ 𝑖𝑚))
6659, 65bitrd 187 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ 𝑖𝑚))
6754, 66mpbird 166 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ⊆ (𝑁𝑚))
68 nnsssuc 6401 . . . . . . . . . . . . . 14 (((𝑁𝑖) ∈ ω ∧ (𝑁𝑚) ∈ ω) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁𝑖) ∈ suc (𝑁𝑚)))
6957, 58, 68syl2anc 408 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ((𝑁𝑖) ⊆ (𝑁𝑚) ↔ (𝑁𝑖) ∈ suc (𝑁𝑚)))
7067, 69mpbid 146 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ suc (𝑁𝑚))
711ad3antrrr 483 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → 𝐹:ω–onto𝐴)
72 fof 5348 . . . . . . . . . . . . . . 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 5279 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → Fun 𝐹)
7573fdmd 5282 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → dom 𝐹 = ω)
7657, 75eleqtrrd 2219 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝑁𝑖) ∈ dom 𝐹)
77 funfvima 5652 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑁𝑖) ∈ dom 𝐹) → ((𝑁𝑖) ∈ suc (𝑁𝑚) → (𝐹‘(𝑁𝑖)) ∈ (𝐹 “ suc (𝑁𝑚))))
7874, 76, 77syl2anc 408 . . . . . . . . . . . 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 2216 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺𝑖) ∈ (𝐹 “ suc (𝑁𝑚)))
8180adantr 274 . . . . . . . . 9 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺𝑖) ∈ (𝐹 “ suc (𝑁𝑚)))
8246, 81eqeltrd 2216 . . . . . . . 8 (((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) ∧ (𝐺‘(𝑁𝑘)) = (𝐺𝑖)) → (𝐺‘(𝑁𝑘)) ∈ (𝐹 “ suc (𝑁𝑚)))
8345, 82mtand 654 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → ¬ (𝐺‘(𝑁𝑘)) = (𝐺𝑖))
8483neqned 2315 . . . . . 6 ((((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) ∧ 𝑖 ∈ (0...𝑚)) → (𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖))
8584ralrimiva 2505 . . . . 5 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖))
86 fveq2 5424 . . . . . . . 8 (𝑗 = (𝑁𝑘) → (𝐺𝑗) = (𝐺‘(𝑁𝑘)))
8786neeq1d 2326 . . . . . . 7 (𝑗 = (𝑁𝑘) → ((𝐺𝑗) ≠ (𝐺𝑖) ↔ (𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)))
8887ralbidv 2437 . . . . . 6 (𝑗 = (𝑁𝑘) → (∀𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖) ↔ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)))
8988rspcev 2789 . . . . 5 (((𝑁𝑘) ∈ ℕ0 ∧ ∀𝑖 ∈ (0...𝑚)(𝐺‘(𝑁𝑘)) ≠ (𝐺𝑖)) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9032, 85, 89syl2anc 408 . . . 4 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑘 ∈ ω ∧ ¬ (𝐹𝑘) ∈ (𝐹 “ suc (𝑁𝑚)))) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9127, 90rexlimddv 2554 . . 3 ((𝜑𝑚 ∈ ℕ0) → ∃𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9291ralrimiva 2505 . 2 (𝜑 → ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖))
9313, 92jca 304 1 (𝜑 → (𝐺:ℕ0onto𝐴 ∧ ∀𝑚 ∈ ℕ0𝑗 ∈ ℕ0𝑖 ∈ (0...𝑚)(𝐺𝑗) ≠ (𝐺𝑖)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071   class class class wbr 3932   ↦ cmpt 3992  suc csuc 4290  ωcom 4507  ◡ccnv 4541  dom cdm 4542   “ cima 4545   ∘ ccom 4546  Fun wfun 5120  ⟶wf 5122  –onto→wfo 5124  –1-1-onto→wf1o 5125  ‘cfv 5126  (class class class)co 5777  freccfrec 6290  0cc0 7639  1c1 7640   + caddc 7642   ≤ cle 7820  ℕ0cn0 8996  ℤcz 9073  ...cfz 9814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-addcom 7739  ax-addass 7741  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-0id 7747  ax-rnegex 7748  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-ltadd 7755 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-recs 6205  df-frec 6291  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-inn 8740  df-n0 8997  df-z 9074  df-uz 9346  df-fz 9815 This theorem is referenced by:  ctinfom  11964
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