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Theorem fargshiftf1 26165
 Description: If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
Hypothesis
Ref Expression
fargshift.g 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
Assertion
Ref Expression
fargshiftf1 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐸
Allowed substitution hints:   𝐺(𝑥)   𝑁(𝑥)

Proof of Theorem fargshiftf1
Dummy variables 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6014 . . 3 (𝐹:(1...𝑁)–1-1→dom 𝐸𝐹:(1...𝑁)⟶dom 𝐸)
2 fargshift.g . . . 4 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
32fargshiftf 26164 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
41, 3sylan2 490 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
5 ffn 5958 . . . . 5 (𝐹:(1...𝑁)⟶dom 𝐸𝐹 Fn (1...𝑁))
6 fseq1hash 13026 . . . . 5 ((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (#‘𝐹) = 𝑁)
75, 6sylan2 490 . . . 4 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (#‘𝐹) = 𝑁)
81, 7sylan2 490 . . 3 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → (#‘𝐹) = 𝑁)
9 eleq1 2676 . . . . 5 ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ0𝑁 ∈ ℕ0))
10 oveq2 6557 . . . . . 6 ((#‘𝐹) = 𝑁 → (1...(#‘𝐹)) = (1...𝑁))
11 f1eq2 6010 . . . . . 6 ((1...(#‘𝐹)) = (1...𝑁) → (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸𝐹:(1...𝑁)–1-1→dom 𝐸))
1210, 11syl 17 . . . . 5 ((#‘𝐹) = 𝑁 → (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸𝐹:(1...𝑁)–1-1→dom 𝐸))
139, 12anbi12d 743 . . . 4 ((#‘𝐹) = 𝑁 → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸)))
14 dff13 6416 . . . . . 6 (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)))
15 fargshiftlem 26162 . . . . . . . . . . 11 (((#‘𝐹) ∈ ℕ0𝑦 ∈ (0..^(#‘𝐹))) → (𝑦 + 1) ∈ (1...(#‘𝐹)))
16 fargshiftlem 26162 . . . . . . . . . . 11 (((#‘𝐹) ∈ ℕ0𝑧 ∈ (0..^(#‘𝐹))) → (𝑧 + 1) ∈ (1...(#‘𝐹)))
1715, 16anim12dan 878 . . . . . . . . . 10 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))))
18 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑦 + 1) → (𝐹𝑘) = (𝐹‘(𝑦 + 1)))
1918eqeq1d 2612 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑦 + 1) → ((𝐹𝑘) = (𝐹𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹𝑙)))
20 eqeq1 2614 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
2119, 20imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑦 + 1) → (((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹𝑙) → (𝑦 + 1) = 𝑙)))
22 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑙 = (𝑧 + 1) → (𝐹𝑙) = (𝐹‘(𝑧 + 1)))
2322eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
2523, 24imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑙 = (𝑧 + 1) → (((𝐹‘(𝑦 + 1)) = (𝐹𝑙) → (𝑦 + 1) = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
2621, 25rspc2v 3293 . . . . . . . . . . . . . . . 16 (((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))) → (∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
2726adantl 481 . . . . . . . . . . . . . . 15 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → (∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
282fargshiftfv 26163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(#‘𝐹)) → (𝐺𝑦) = (𝐹‘(𝑦 + 1))))
2928expcom 450 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ0 → (𝑦 ∈ (0..^(#‘𝐹)) → (𝐺𝑦) = (𝐹‘(𝑦 + 1)))))
3029com13 86 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0..^(#‘𝐹)) → ((#‘𝐹) ∈ ℕ0 → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑦) = (𝐹‘(𝑦 + 1)))))
3130adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → ((#‘𝐹) ∈ ℕ0 → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑦) = (𝐹‘(𝑦 + 1)))))
3231impcom 445 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑦) = (𝐹‘(𝑦 + 1))))
3332impcom 445 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → (𝐺𝑦) = (𝐹‘(𝑦 + 1)))
342fargshiftfv 26163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(#‘𝐹)) → (𝐺𝑧) = (𝐹‘(𝑧 + 1))))
3534expcom 450 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ0 → (𝑧 ∈ (0..^(#‘𝐹)) → (𝐺𝑧) = (𝐹‘(𝑧 + 1)))))
3635com13 86 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ (0..^(#‘𝐹)) → ((#‘𝐹) ∈ ℕ0 → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑧) = (𝐹‘(𝑧 + 1)))))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → ((#‘𝐹) ∈ ℕ0 → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑧) = (𝐹‘(𝑧 + 1)))))
3837impcom 445 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺𝑧) = (𝐹‘(𝑧 + 1))))
3938impcom 445 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → (𝐺𝑧) = (𝐹‘(𝑧 + 1)))
4033, 39eqeq12d 2625 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → ((𝐺𝑦) = (𝐺𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
4140adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → ((𝐺𝑦) = (𝐺𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
4241adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐺𝑦) = (𝐺𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
43 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℤ)
4443zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℂ)
4544adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → 𝑦 ∈ ℂ)
4645adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → 𝑦 ∈ ℂ)
47 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ (0..^(#‘𝐹)) → 𝑧 ∈ ℤ)
4847zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (0..^(#‘𝐹)) → 𝑧 ∈ ℂ)
4948adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → 𝑧 ∈ ℂ)
5049adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → 𝑧 ∈ ℂ)
51 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . 23 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → 1 ∈ ℂ)
5246, 50, 513jca 1235 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
5352adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
5453adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
55 addcan2 10100 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) = (𝑧 + 1) ↔ 𝑦 = 𝑧))
5654, 55syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → ((𝑦 + 1) = (𝑧 + 1) ↔ 𝑦 = 𝑧))
5756imbi2d 329 . . . . . . . . . . . . . . . . . 18 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → (((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1)) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → 𝑦 = 𝑧)))
5857biimpa 500 . . . . . . . . . . . . . . . . 17 ((((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → 𝑦 = 𝑧))
5942, 58sylbid 229 . . . . . . . . . . . . . . . 16 ((((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))
6059ex 449 . . . . . . . . . . . . . . 15 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → (((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1)) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
6127, 60syld 46 . . . . . . . . . . . . . 14 (((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹)))) → (∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
6261exp31 628 . . . . . . . . . . . . 13 (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))) → (∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))))
6362com24 93 . . . . . . . . . . . 12 (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙) → (((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))))
6463imp 444 . . . . . . . . . . 11 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → (((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))))
6564com13 86 . . . . . . . . . 10 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))) → ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))))
6617, 65mpd 15 . . . . . . . . 9 (((#‘𝐹) ∈ ℕ0 ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
6766expcom 450 . . . . . . . 8 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → ((#‘𝐹) ∈ ℕ0 → ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))))
6867com13 86 . . . . . . 7 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → ((#‘𝐹) ∈ ℕ0 → ((𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))) → ((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))))
6968ralrimdvv 2956 . . . . . 6 ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(#‘𝐹))∀𝑙 ∈ (1...(#‘𝐹))((𝐹𝑘) = (𝐹𝑙) → 𝑘 = 𝑙)) → ((#‘𝐹) ∈ ℕ0 → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
7014, 69sylbi 206 . . . . 5 (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸 → ((#‘𝐹) ∈ ℕ0 → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
7170impcom 445 . . . 4 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))
7213, 71syl6bir 243 . . 3 ((#‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
738, 72mpcom 37 . 2 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧))
74 dff13 6416 . 2 (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺𝑦) = (𝐺𝑧) → 𝑦 = 𝑧)))
754, 73, 74sylanbrc 695 1 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980 This theorem is referenced by:  eupatrl  26495
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