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Theorem fseq1p1m1 13607
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 1191 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2 nn0p1nn 12541 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
32adantr 479 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℕ)
4 simpr2 1192 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐵𝐴)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6 fsng 7142 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
75, 6mpbiri 257 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
83, 4, 7syl2anc 582 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
94snssd 4808 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → {𝐵} ⊆ 𝐴)
108, 9fssd 6735 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11 fzp1disj 13592 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 11 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
131, 10, 12fun2d 6756 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14 1z 12622 . . . . . . . 8 1 ∈ ℤ
15 simpl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℕ0)
16 nn0uz 12894 . . . . . . . . . 10 0 = (ℤ‘0)
17 1m1e0 12314 . . . . . . . . . . 11 (1 − 1) = 0
1817fveq2i 6895 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
1916, 18eqtr4i 2756 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2015, 19eleqtrdi 2835 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ (ℤ‘(1 − 1)))
21 fzsuc2 13591 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2214, 20, 21sylancr 585 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2322eqcomd 2731 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2423feq2d 6703 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2513, 24mpbid 231 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴)
26 simpr3 1193 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2726feq1d 6702 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2825, 27mpbird 256 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
29 ovex 7449 . . . . . 6 (𝑁 + 1) ∈ V
3029snid 4660 . . . . 5 (𝑁 + 1) ∈ {(𝑁 + 1)}
31 fvres 6911 . . . . 5 ((𝑁 + 1) ∈ {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
3230, 31ax-mp 5 . . . 4 ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1))
3326reseq1d 5978 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
34 ffn 6717 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶𝐴𝐹 Fn (1...𝑁))
35 fnresdisj 6670 . . . . . . . . . . 11 (𝐹 Fn (1...𝑁) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
361, 34, 353syl 18 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
3712, 36mpbid 231 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3837uneq1d 4155 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
39 resundir 5994 . . . . . . . 8 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
40 uncom 4146 . . . . . . . . 9 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
41 un0 4386 . . . . . . . . 9 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
4240, 41eqtr2i 2754 . . . . . . . 8 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4338, 39, 423eqtr4g 2790 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
44 ffn 6717 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶𝐴𝐻 Fn {(𝑁 + 1)})
45 fnresdm 6669 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4610, 44, 453syl 18 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4733, 43, 463eqtrd 2769 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4847fveq1d 6894 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
495fveq1i 6893 . . . . . . 7 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
50 fvsng 7185 . . . . . . 7 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
5149, 50eqtrid 2777 . . . . . 6 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
523, 4, 51syl2anc 582 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
5348, 52eqtrd 2765 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = 𝐵)
5432, 53eqtr3id 2779 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
5526reseq1d 5978 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
56 incom 4195 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5756, 12eqtrid 2777 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
58 ffn 6717 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
59 fnresdisj 6670 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
608, 58, 593syl 18 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
6157, 60mpbid 231 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6261uneq2d 4156 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
63 resundir 5994 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
64 un0 4386 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6564eqcomi 2734 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6662, 63, 653eqtr4g 2790 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
67 fnresdm 6669 . . . . 5 (𝐹 Fn (1...𝑁) → (𝐹 ↾ (1...𝑁)) = 𝐹)
681, 34, 673syl 18 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
6955, 66, 683eqtrrd 2770 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7028, 54, 693jca 1125 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁))))
71 simpr1 1191 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
72 fzssp1 13576 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
73 fssres 6758 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
7471, 72, 73sylancl 584 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
75 simpr3 1193 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7675feq1d 6702 . . . 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))
7774, 76mpbird 256 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
78 simpr2 1192 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
792adantr 479 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
80 nnuz 12895 . . . . . . 7 ℕ = (ℤ‘1)
8179, 80eleqtrdi 2835 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ‘1))
82 eluzfz2 13541 . . . . . 6 ((𝑁 + 1) ∈ (ℤ‘1) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8381, 82syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8471, 83ffvelcdmd 7090 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
8578, 84eqeltrrd 2826 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵𝐴)
86 ffn 6717 . . . . . . . . 9 (𝐺:(1...(𝑁 + 1))⟶𝐴𝐺 Fn (1...(𝑁 + 1)))
8771, 86syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 Fn (1...(𝑁 + 1)))
88 fnressn 7163 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
8987, 83, 88syl2anc 582 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
90 opeq2 4870 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
9190sneqd 4636 . . . . . . . 8 ((𝐺‘(𝑁 + 1)) = 𝐵 → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
9278, 91syl 17 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
9389, 92eqtrd 2765 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
945, 93eqtr4id 2784 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9575, 94uneq12d 4157 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
96 simpl 481 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
9796, 19eleqtrdi 2835 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ‘(1 − 1)))
9814, 97, 21sylancr 585 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
9998reseq2d 5979 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})))
100 resundi 5993 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
10199, 100eqtr2di 2782 . . . 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))))
102 fnresdm 6669 . . . . 5 (𝐺 Fn (1...(𝑁 + 1)) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10371, 86, 1023syl 18 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10495, 101, 1033eqtrrd 2770 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10577, 85, 1043jca 1125 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)))
10670, 105impbida 799 1 (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cun 3937  cin 3938  wss 3939  c0 4318  {csn 4624  cop 4630  cres 5674   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7416  0cc0 11138  1c1 11139   + caddc 11141  cmin 11474  cn 12242  0cn0 12502  cz 12588  cuz 12852  ...cfz 13516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517
This theorem is referenced by:  fseq1m1p1  13608
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