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Theorem fseq1p1m1 12621
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 1248 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2 nn0p1nn 11579 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
32adantr 472 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝑁 + 1) ∈ ℕ)
4 simpr2 1250 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐵𝐴)
5 fseq1p1m1.1 . . . . . . . . 9 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6 fsng 6595 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
75, 6mpbiri 249 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
83, 4, 7syl2anc 579 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
94snssd 4494 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → {𝐵} ⊆ 𝐴)
108, 9fssd 6237 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11 fzp1disj 12606 . . . . . . 7 ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
1211a1i 11 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13 fun2 6249 . . . . . 6 (((𝐹:(1...𝑁)⟶𝐴𝐻:{(𝑁 + 1)}⟶𝐴) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
141, 10, 12, 13syl21anc 866 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
15 1z 11654 . . . . . . . 8 1 ∈ ℤ
16 simpl 474 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ ℕ0)
17 nn0uz 11922 . . . . . . . . . 10 0 = (ℤ‘0)
18 1m1e0 11344 . . . . . . . . . . 11 (1 − 1) = 0
1918fveq2i 6378 . . . . . . . . . 10 (ℤ‘(1 − 1)) = (ℤ‘0)
2017, 19eqtr4i 2790 . . . . . . . . 9 0 = (ℤ‘(1 − 1))
2116, 20syl6eleq 2854 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝑁 ∈ (ℤ‘(1 − 1)))
22 fzsuc2 12605 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2315, 21, 22sylancr 581 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
2423eqcomd 2771 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
2524feq2d 6209 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2614, 25mpbid 223 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴)
27 simpr3 1252 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺 = (𝐹𝐻))
2827feq1d 6208 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹𝐻):(1...(𝑁 + 1))⟶𝐴))
2926, 28mpbird 248 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
30 ovex 6874 . . . . . 6 (𝑁 + 1) ∈ V
3130snid 4366 . . . . 5 (𝑁 + 1) ∈ {(𝑁 + 1)}
32 fvres 6394 . . . . 5 ((𝑁 + 1) ∈ {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
3331, 32ax-mp 5 . . . 4 ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1))
3427reseq1d 5564 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹𝐻) ↾ {(𝑁 + 1)}))
35 ffn 6223 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶𝐴𝐹 Fn (1...𝑁))
36 fnresdisj 6179 . . . . . . . . . . 11 (𝐹 Fn (1...𝑁) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
371, 35, 363syl 18 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
3812, 37mpbid 223 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
3938uneq1d 3928 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
40 resundir 5587 . . . . . . . 8 ((𝐹𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
41 uncom 3919 . . . . . . . . 9 (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
42 un0 4129 . . . . . . . . 9 ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
4341, 42eqtr2i 2788 . . . . . . . 8 (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
4439, 40, 433eqtr4g 2824 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
45 ffn 6223 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶𝐴𝐻 Fn {(𝑁 + 1)})
46 fnresdm 6178 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4710, 45, 463syl 18 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
4834, 44, 473eqtrd 2803 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
4948fveq1d 6377 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
505fveq1i 6376 . . . . . . 7 (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
51 fvsng 6640 . . . . . . 7 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
5250, 51syl5eq 2811 . . . . . 6 (((𝑁 + 1) ∈ ℕ ∧ 𝐵𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
533, 4, 52syl2anc 579 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
5449, 53eqtrd 2799 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = 𝐵)
5533, 54syl5eqr 2813 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
5627reseq1d 5564 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹𝐻) ↾ (1...𝑁)))
57 incom 3967 . . . . . . . 8 ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
5857, 12syl5eq 2811 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
59 ffn 6223 . . . . . . . 8 (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
60 fnresdisj 6179 . . . . . . . 8 (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
618, 59, 603syl 18 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
6258, 61mpbid 223 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
6362uneq2d 3929 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
64 resundir 5587 . . . . 5 ((𝐹𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
65 un0 4129 . . . . . 6 ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
6665eqcomi 2774 . . . . 5 (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
6763, 64, 663eqtr4g 2824 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → ((𝐹𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
68 fnresdm 6178 . . . . 5 (𝐹 Fn (1...𝑁) → (𝐹 ↾ (1...𝑁)) = 𝐹)
691, 35, 683syl 18 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
7056, 67, 693eqtrrd 2804 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7129, 55, 703jca 1158 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁))))
72 simpr1 1248 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
73 fzssp1 12591 . . . . 5 (1...𝑁) ⊆ (1...(𝑁 + 1))
74 fssres 6252 . . . . 5 ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
7572, 73, 74sylancl 580 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
76 simpr3 1252 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
7776feq1d 6208 . . . 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 248 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
79 simpr2 1250 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
802adantr 472 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
81 nnuz 11923 . . . . . . 7 ℕ = (ℤ‘1)
8280, 81syl6eleq 2854 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ‘1))
83 eluzfz2 12556 . . . . . 6 ((𝑁 + 1) ∈ (ℤ‘1) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8482, 83syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
8572, 84ffvelrnd 6550 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
8679, 85eqeltrrd 2845 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵𝐴)
87 ffn 6223 . . . . . . . . 9 (𝐺:(1...(𝑁 + 1))⟶𝐴𝐺 Fn (1...(𝑁 + 1)))
8872, 87syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 Fn (1...(𝑁 + 1)))
89 fnressn 6617 . . . . . . . 8 ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
9088, 84, 89syl2anc 579 . . . . . . 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 4560 . . . . . . . . 9 ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
9291sneqd 4346 . . . . . . . 8 ((𝐺‘(𝑁 + 1)) = 𝐵 → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
9379, 92syl 17 . . . . . . 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 2799 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
9594, 5syl6reqr 2818 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
9676, 95uneq12d 3930 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
97 simpl 474 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
9897, 20syl6eleq 2854 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ‘(1 − 1)))
9915, 98, 22sylancr 581 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
10099reseq2d 5565 . . . . 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 5586 . . . . 5 (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
102100, 101syl6req 2816 . . . 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 6178 . . . . 5 (𝐺 Fn (1...(𝑁 + 1)) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10472, 87, 1033syl 18 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
10596, 102, 1043eqtrrd 2804 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹𝐻))
10678, 86, 1053jca 1158 . 2 ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)))
10771, 106impbida 835 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340  cres 5279   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  cmin 10520  cn 11274  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534
This theorem is referenced by:  fseq1m1p1  12622
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