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Theorem fseq1p1m1 10126
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 1005 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2 nn0p1nn 9246 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
32adantr 276 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℕ)
4 simpr2 1006 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐵𝐴)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6 fsng 5710 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
75, 6mpbiri 168 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
83, 4, 7syl2anc 411 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
94snssd 3752 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → {𝐵} ⊆ 𝐴)
108, 9fssd 5397 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11 fzp1disj 10112 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 9 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13 fun2 5408 . . . . . 6 (((𝐹:(1...𝑁)⟶𝐴𝐻:{(𝑁 + 1)}⟶𝐴) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
141, 10, 12, 13syl21anc 1248 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
15 1z 9310 . . . . . . . 8 1 ∈ ℤ
16 simpl 109 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℕ0)
17 nn0uz 9594 . . . . . . . . . 10 0 = (ℤ‘0)
18 1m1e0 9019 . . . . . . . . . . 11 (1 − 1) = 0
1918fveq2i 5537 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
2017, 19eqtr4i 2213 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2116, 20eleqtrdi 2282 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ (ℤ‘(1 − 1)))
22 fzsuc2 10111 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2315, 21, 22sylancr 414 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2423eqcomd 2195 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2524feq2d 5372 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2614, 25mpbid 147 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴)
27 simpr3 1007 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2827feq1d 5371 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2926, 28mpbird 167 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
3027reseq1d 4924 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
31 ffn 5384 . . . . . . . . . 10 (𝐹:(1...𝑁)⟶𝐴𝐹 Fn (1...𝑁))
32 fnresdisj 5345 . . . . . . . . . 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 147 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3534uneq1d 3303 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
36 resundir 4939 . . . . . . 7 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
37 uncom 3294 . . . . . . . 8 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
38 un0 3471 . . . . . . . 8 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
3937, 38eqtr2i 2211 . . . . . . 7 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4035, 36, 393eqtr4g 2247 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
41 ffn 5384 . . . . . . 7 (𝐻:{(𝑁 + 1)}⟶𝐴𝐻 Fn {(𝑁 + 1)})
42 fnresdm 5344 . . . . . . 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 2226 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4544fveq1d 5536 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
4616nn0zd 9404 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℤ)
4746peano2zd 9409 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℤ)
48 snidg 3636 . . . . 5 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ {(𝑁 + 1)})
49 fvres 5558 . . . . 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 5535 . . . . . 6 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
52 fvsng 5733 . . . . . 6 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
5351, 52eqtrid 2234 . . . . 5 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
543, 4, 53syl2anc 411 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
5545, 50, 543eqtr3d 2230 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
5627reseq1d 4924 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
57 incom 3342 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5857, 12eqtrid 2234 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
59 ffn 5384 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
60 fnresdisj 5345 . . . . . . . 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 147 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6362uneq2d 3304 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
64 resundir 4939 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
65 un0 3471 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6665eqcomi 2193 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6763, 64, 663eqtr4g 2247 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
68 fnresdm 5344 . . . . 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 2227 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7129, 55, 703jca 1179 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁))))
72 simpr1 1005 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
73 fzssp1 10099 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
74 fssres 5410 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
7572, 73, 74sylancl 413 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
76 simpr3 1007 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7776feq1d 5371 . . . 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 167 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
79 simpr2 1006 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
802adantr 276 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
81 nnuz 9595 . . . . . . 7 ℕ = (ℤ‘1)
8280, 81eleqtrdi 2282 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ‘1))
83 eluzfz2 10064 . . . . . 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, 84ffvelcdmd 5673 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
8679, 85eqeltrrd 2267 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵𝐴)
87 ffn 5384 . . . . . . . . 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 5723 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
9088, 84, 89syl2anc 411 . . . . . . 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 3794 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
9291sneqd 3620 . . . . . . . 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 2222 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
955, 94eqtr4id 2241 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9676, 95uneq12d 3305 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
97 simpl 109 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
9897, 20eleqtrdi 2282 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ‘(1 − 1)))
9915, 98, 22sylancr 414 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
10099reseq2d 4925 . . . . 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 4938 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
102100, 101eqtr2di 2239 . . . 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 5344 . . . . 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 2227 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10678, 86, 1053jca 1179 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)))
10771, 106impbida 596 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 104  wb 105  w3a 980   = wceq 1364  wcel 2160  cun 3142  cin 3143  wss 3144  c0 3437  {csn 3607  cop 3610  cres 4646   Fn wfn 5230  wf 5231  cfv 5235  (class class class)co 5897  0cc0 7842  1c1 7843   + caddc 7845  cmin 8159  cn 8950  0cn0 9207  cz 9284  cuz 9559  ...cfz 10040
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-fz 10041
This theorem is referenced by:  fseq1m1p1  10127
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