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Theorem fseq1p1m1 8726
 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 0 → ((𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))))

Proof of Theorem fseq1p1m1
StepHypRef Expression
1 simpr1 909 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶A)
2 nn0p1nn 7997 . . . . . . . . 9 (𝑁 0 → (𝑁 + 1) ℕ)
32adantr 261 . . . . . . . 8 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝑁 + 1) ℕ)
4 simpr2 910 . . . . . . . 8 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → B A)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), B⟩}
6 fsng 5279 . . . . . . . . 9 (((𝑁 + 1) B A) → (𝐻:{(𝑁 + 1)}⟶{B} ↔ 𝐻 = {⟨(𝑁 + 1), B⟩}))
75, 6mpbiri 157 . . . . . . . 8 (((𝑁 + 1) B A) → 𝐻:{(𝑁 + 1)}⟶{B})
83, 4, 7syl2anc 391 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{B})
94snssd 3500 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → {B} ⊆ A)
108, 9fssd 4998 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶A)
11 fzp1disj 8712 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 9 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13 fun2 5007 . . . . . 6 (((𝐹:(1...𝑁)⟶A 𝐻:{(𝑁 + 1)}⟶A) ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶A)
141, 10, 12, 13syl21anc 1133 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶A)
15 1z 8047 . . . . . . . 8 1
16 simpl 102 . . . . . . . . 9 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝑁 0)
17 nn0uz 8283 . . . . . . . . . 10 0 = (ℤ‘0)
18 1m1e0 7764 . . . . . . . . . . 11 (1 − 1) = 0
1918fveq2i 5124 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
2017, 19eqtr4i 2060 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2116, 20syl6eleq 2127 . . . . . . . 8 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝑁 (ℤ‘(1 − 1)))
22 fzsuc2 8711 . . . . . . . 8 ((1 𝑁 (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2315, 21, 22sylancr 393 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2423eqcomd 2042 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2524feq2d 4978 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶A ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶A))
2614, 25mpbid 135 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶A)
27 simpr3 911 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2827feq1d 4977 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶A ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶A))
2926, 28mpbird 156 . . 3 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶A)
3027reseq1d 4554 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
31 ffn 4989 . . . . . . . . . 10 (𝐹:(1...𝑁)⟶A𝐹 Fn (1...𝑁))
32 fnresdisj 4952 . . . . . . . . . 10 (𝐹 Fn (1...𝑁) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
331, 31, 323syl 17 . . . . . . . . 9 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
3412, 33mpbid 135 . . . . . . . 8 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3534uneq1d 3090 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
36 resundir 4569 . . . . . . 7 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
37 uncom 3081 . . . . . . . 8 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
38 un0 3245 . . . . . . . 8 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
3937, 38eqtr2i 2058 . . . . . . 7 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4035, 36, 393eqtr4g 2094 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
41 ffn 4989 . . . . . . 7 (𝐻:{(𝑁 + 1)}⟶A𝐻 Fn {(𝑁 + 1)})
42 fnresdm 4951 . . . . . . 7 (𝐻 Fn {(𝑁 + 1)} → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4310, 41, 423syl 17 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4430, 40, 433eqtrd 2073 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4544fveq1d 5123 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
4616nn0zd 8134 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝑁 ℤ)
4746peano2zd 8139 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝑁 + 1) ℤ)
48 snidg 3392 . . . . 5 ((𝑁 + 1) ℤ → (𝑁 + 1) {(𝑁 + 1)})
49 fvres 5141 . . . . 5 ((𝑁 + 1) {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
5047, 48, 493syl 17 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
515fveq1i 5122 . . . . . 6 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), B⟩}‘(𝑁 + 1))
52 fvsng 5302 . . . . . 6 (((𝑁 + 1) B A) → ({⟨(𝑁 + 1), B⟩}‘(𝑁 + 1)) = B)
5351, 52syl5eq 2081 . . . . 5 (((𝑁 + 1) B A) → (𝐻‘(𝑁 + 1)) = B)
543, 4, 53syl2anc 391 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = B)
5545, 50, 543eqtr3d 2077 . . 3 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = B)
5627reseq1d 4554 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
57 incom 3123 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5857, 12syl5eq 2081 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
59 ffn 4989 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{B} → 𝐻 Fn {(𝑁 + 1)})
60 fnresdisj 4952 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
618, 59, 603syl 17 . . . . . . 7 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
6258, 61mpbid 135 . . . . . 6 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6362uneq2d 3091 . . . . 5 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
64 resundir 4569 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
65 un0 3245 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6665eqcomi 2041 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6763, 64, 663eqtr4g 2094 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
68 fnresdm 4951 . . . . 5 (𝐹 Fn (1...𝑁) → (𝐹 ↾ (1...𝑁)) = 𝐹)
691, 31, 683syl 17 . . . 4 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
7056, 67, 693eqtrrd 2074 . . 3 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7129, 55, 703jca 1083 . 2 ((𝑁 0 (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁))))
72 simpr1 909 . . . . 5 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶A)
73 fzssp1 8700 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
74 fssres 5009 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶A (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶A)
7572, 73, 74sylancl 392 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶A)
76 simpr3 911 . . . . 5 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7776feq1d 4977 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶A ↔ (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶A))
7875, 77mpbird 156 . . 3 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶A)
79 simpr2 910 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = B)
802adantr 261 . . . . . . 7 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ℕ)
81 nnuz 8284 . . . . . . 7 ℕ = (ℤ‘1)
8280, 81syl6eleq 2127 . . . . . 6 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) (ℤ‘1))
83 eluzfz2 8666 . . . . . 6 ((𝑁 + 1) (ℤ‘1) → (𝑁 + 1) (1...(𝑁 + 1)))
8482, 83syl 14 . . . . 5 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) (1...(𝑁 + 1)))
8572, 84ffvelrnd 5246 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) A)
8679, 85eqeltrrd 2112 . . 3 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → B A)
87 ffn 4989 . . . . . . . . 9 (𝐺:(1...(𝑁 + 1))⟶A𝐺 Fn (1...(𝑁 + 1)))
8872, 87syl 14 . . . . . . . 8 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 Fn (1...(𝑁 + 1)))
89 fnressn 5292 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) (𝑁 + 1) (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
9088, 84, 89syl2anc 391 . . . . . . 7 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
91 opeq2 3541 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = B → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), B⟩)
9291sneqd 3380 . . . . . . . 8 ((𝐺‘(𝑁 + 1)) = B → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), B⟩})
9379, 92syl 14 . . . . . . 7 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), B⟩})
9490, 93eqtrd 2069 . . . . . 6 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), B⟩})
9594, 5syl6reqr 2088 . . . . 5 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9676, 95uneq12d 3092 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
97 simpl 102 . . . . . . . 8 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 0)
9897, 20syl6eleq 2127 . . . . . . 7 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 (ℤ‘(1 − 1)))
9915, 98, 22sylancr 393 . . . . . 6 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
10099reseq2d 4555 . . . . 5 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})))
101 resundi 4568 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
102100, 101syl6req 2086 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})) = (𝐺 ↾ (1...(𝑁 + 1))))
103 fnresdm 4951 . . . . 5 (𝐺 Fn (1...(𝑁 + 1)) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10472, 87, 1033syl 17 . . . 4 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10596, 102, 1043eqtrrd 2074 . . 3 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10678, 86, 1053jca 1083 . 2 ((𝑁 0 (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻)))
10771, 106impbida 528 1 (𝑁 0 → ((𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  {csn 3367  ⟨cop 3370   ↾ cres 4290   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  0cc0 6711  1c1 6712   + caddc 6714   − cmin 6979  ℕcn 7695  ℕ0cn0 7957  ℤcz 8021  ℤ≥cuz 8249  ...cfz 8644 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250  df-fz 8645 This theorem is referenced by:  fseq1m1p1  8727
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