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Theorem ctinfomlemom 11963
Description: Lemma for ctinfom 11964. 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 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  ontowfo 5124  1-1-ontowf1o 5125  cfv 5126  (class class class)co 5777  freccfrec 6290  0cc0 7639  1c1 7640   + caddc 7642  cle 7820  0cn0 8996  cz 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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