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Theorem fseq1p1m1 13638
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 1195 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2 nn0p1nn 12565 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
32adantr 480 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℕ)
4 simpr2 1196 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐵𝐴)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6 fsng 7157 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
75, 6mpbiri 258 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
83, 4, 7syl2anc 584 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
94snssd 4809 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → {𝐵} ⊆ 𝐴)
108, 9fssd 6753 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11 fzp1disj 13623 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 11 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
131, 10, 12fun2d 6772 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14 1z 12647 . . . . . . . 8 1 ∈ ℤ
15 simpl 482 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℕ0)
16 nn0uz 12920 . . . . . . . . . 10 0 = (ℤ‘0)
17 1m1e0 12338 . . . . . . . . . . 11 (1 − 1) = 0
1817fveq2i 6909 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
1916, 18eqtr4i 2768 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2015, 19eleqtrdi 2851 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ (ℤ‘(1 − 1)))
21 fzsuc2 13622 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2214, 20, 21sylancr 587 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2322eqcomd 2743 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2423feq2d 6722 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2513, 24mpbid 232 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴)
26 simpr3 1197 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2726feq1d 6720 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2825, 27mpbird 257 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
29 ovex 7464 . . . . . 6 (𝑁 + 1) ∈ V
3029snid 4662 . . . . 5 (𝑁 + 1) ∈ {(𝑁 + 1)}
31 fvres 6925 . . . . 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 5996 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
34 ffn 6736 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶𝐴𝐹 Fn (1...𝑁))
35 fnresdisj 6688 . . . . . . . . . . 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 232 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3837uneq1d 4167 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
39 resundir 6012 . . . . . . . 8 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
40 uncom 4158 . . . . . . . . 9 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
41 un0 4394 . . . . . . . . 9 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
4240, 41eqtr2i 2766 . . . . . . . 8 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4338, 39, 423eqtr4g 2802 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
44 ffn 6736 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶𝐴𝐻 Fn {(𝑁 + 1)})
45 fnresdm 6687 . . . . . . . 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 2781 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4847fveq1d 6908 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
495fveq1i 6907 . . . . . . 7 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
50 fvsng 7200 . . . . . . 7 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
5149, 50eqtrid 2789 . . . . . 6 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
523, 4, 51syl2anc 584 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
5348, 52eqtrd 2777 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = 𝐵)
5432, 53eqtr3id 2791 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
5526reseq1d 5996 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
56 incom 4209 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5756, 12eqtrid 2789 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
58 ffn 6736 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
59 fnresdisj 6688 . . . . . . . 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 232 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6261uneq2d 4168 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
63 resundir 6012 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
64 un0 4394 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6564eqcomi 2746 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6662, 63, 653eqtr4g 2802 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
67 fnresdm 6687 . . . . 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 2782 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7028, 54, 693jca 1129 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁))))
71 simpr1 1195 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
72 fzssp1 13607 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
73 fssres 6774 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
7471, 72, 73sylancl 586 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
75 simpr3 1197 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7675feq1d 6720 . . . 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 257 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
78 simpr2 1196 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
792adantr 480 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
80 nnuz 12921 . . . . . . 7 ℕ = (ℤ‘1)
8179, 80eleqtrdi 2851 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ‘1))
82 eluzfz2 13572 . . . . . 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 7105 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
8578, 84eqeltrrd 2842 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵𝐴)
86 ffn 6736 . . . . . . . . 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 7178 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
8987, 83, 88syl2anc 584 . . . . . . 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 4874 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
9190sneqd 4638 . . . . . . . 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 2777 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
945, 93eqtr4id 2796 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9575, 94uneq12d 4169 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
96 simpl 482 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
9796, 19eleqtrdi 2851 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ‘(1 − 1)))
9814, 97, 21sylancr 587 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
9998reseq2d 5997 . . . . 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 6011 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
10199, 100eqtr2di 2794 . . . 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 6687 . . . . 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 2782 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10577, 85, 1043jca 1129 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)))
10670, 105impbida 801 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548
This theorem is referenced by:  fseq1m1p1  13639
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