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Theorem fseq1p1m1 9825
Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
Hypothesis
Ref Expression
fseq1p1m1.1 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
Assertion
Ref Expression
fseq1p1m1 (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))))

Proof of Theorem fseq1p1m1
StepHypRef Expression
1 simpr1 970 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2 nn0p1nn 8970 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
32adantr 272 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℕ)
4 simpr2 971 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐵𝐴)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6 fsng 5559 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
75, 6mpbiri 167 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
83, 4, 7syl2anc 406 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
94snssd 3633 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → {𝐵} ⊆ 𝐴)
108, 9fssd 5253 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11 fzp1disj 9811 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 9 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13 fun2 5264 . . . . . 6 (((𝐹:(1...𝑁)⟶𝐴𝐻:{(𝑁 + 1)}⟶𝐴) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
141, 10, 12, 13syl21anc 1198 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
15 1z 9034 . . . . . . . 8 1 ∈ ℤ
16 simpl 108 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℕ0)
17 nn0uz 9312 . . . . . . . . . 10 0 = (ℤ‘0)
18 1m1e0 8749 . . . . . . . . . . 11 (1 − 1) = 0
1918fveq2i 5390 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
2017, 19eqtr4i 2139 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2116, 20syl6eleq 2208 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ (ℤ‘(1 − 1)))
22 fzsuc2 9810 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2315, 21, 22sylancr 408 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2423eqcomd 2121 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2524feq2d 5228 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2614, 25mpbid 146 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴)
27 simpr3 972 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2827feq1d 5227 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2926, 28mpbird 166 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
3027reseq1d 4786 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
31 ffn 5240 . . . . . . . . . 10 (𝐹:(1...𝑁)⟶𝐴𝐹 Fn (1...𝑁))
32 fnresdisj 5201 . . . . . . . . . 10 (𝐹 Fn (1...𝑁) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
331, 31, 323syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
3412, 33mpbid 146 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3534uneq1d 3197 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
36 resundir 4801 . . . . . . 7 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
37 uncom 3188 . . . . . . . 8 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
38 un0 3364 . . . . . . . 8 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
3937, 38eqtr2i 2137 . . . . . . 7 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4035, 36, 393eqtr4g 2173 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
41 ffn 5240 . . . . . . 7 (𝐻:{(𝑁 + 1)}⟶𝐴𝐻 Fn {(𝑁 + 1)})
42 fnresdm 5200 . . . . . . 7 (𝐻 Fn {(𝑁 + 1)} → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4310, 41, 423syl 17 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4430, 40, 433eqtrd 2152 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4544fveq1d 5389 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
4616nn0zd 9125 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℤ)
4746peano2zd 9130 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℤ)
48 snidg 3522 . . . . 5 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ {(𝑁 + 1)})
49 fvres 5411 . . . . 5 ((𝑁 + 1) ∈ {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
5047, 48, 493syl 17 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
515fveq1i 5388 . . . . . 6 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
52 fvsng 5582 . . . . . 6 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
5351, 52syl5eq 2160 . . . . 5 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
543, 4, 53syl2anc 406 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
5545, 50, 543eqtr3d 2156 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
5627reseq1d 4786 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
57 incom 3236 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5857, 12syl5eq 2160 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
59 ffn 5240 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
60 fnresdisj 5201 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
618, 59, 603syl 17 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
6258, 61mpbid 146 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6362uneq2d 3198 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
64 resundir 4801 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
65 un0 3364 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6665eqcomi 2119 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6763, 64, 663eqtr4g 2173 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
68 fnresdm 5200 . . . . 5 (𝐹 Fn (1...𝑁) → (𝐹 ↾ (1...𝑁)) = 𝐹)
691, 31, 683syl 17 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
7056, 67, 693eqtrrd 2153 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7129, 55, 703jca 1144 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁))))
72 simpr1 970 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
73 fzssp1 9798 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
74 fssres 5266 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
7572, 73, 74sylancl 407 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
76 simpr3 972 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7776feq1d 5227 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴 ↔ (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴))
7875, 77mpbird 166 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
79 simpr2 971 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
802adantr 272 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
81 nnuz 9313 . . . . . . 7 ℕ = (ℤ‘1)
8280, 81syl6eleq 2208 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ‘1))
83 eluzfz2 9763 . . . . . 6 ((𝑁 + 1) ∈ (ℤ‘1) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8482, 83syl 14 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8572, 84ffvelrnd 5522 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
8679, 85eqeltrrd 2193 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵𝐴)
87 ffn 5240 . . . . . . . . 9 (𝐺:(1...(𝑁 + 1))⟶𝐴𝐺 Fn (1...(𝑁 + 1)))
8872, 87syl 14 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 Fn (1...(𝑁 + 1)))
89 fnressn 5572 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
9088, 84, 89syl2anc 406 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
91 opeq2 3674 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
9291sneqd 3508 . . . . . . . 8 ((𝐺‘(𝑁 + 1)) = 𝐵 → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
9379, 92syl 14 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
9490, 93eqtrd 2148 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
9594, 5syl6reqr 2167 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9676, 95uneq12d 3199 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
97 simpl 108 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
9897, 20syl6eleq 2208 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ‘(1 − 1)))
9915, 98, 22sylancr 408 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
10099reseq2d 4787 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})))
101 resundi 4800 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
102100, 101syl6req 2165 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})) = (𝐺 ↾ (1...(𝑁 + 1))))
103 fnresdm 5200 . . . . 5 (𝐺 Fn (1...(𝑁 + 1)) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10472, 87, 1033syl 17 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10596, 102, 1043eqtrrd 2153 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10678, 86, 1053jca 1144 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)))
10771, 106impbida 568 1 (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wcel 1463  cun 3037  cin 3038  wss 3039  c0 3331  {csn 3495  cop 3498  cres 4509   Fn wfn 5086  wf 5087  cfv 5091  (class class class)co 5740  0cc0 7584  1c1 7585   + caddc 7587  cmin 7897  cn 8680  0cn0 8931  cz 9008  cuz 9278  ...cfz 9741
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742
This theorem is referenced by:  fseq1m1p1  9826
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