| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fof 6819 | . . 3
⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → 𝐹:(1...𝑁)⟶dom 𝐸) | 
| 2 |  | fargshift.g | . . . 4
⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) | 
| 3 | 2 | fargshiftf 47432 | . . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) | 
| 4 | 1, 3 | sylan2 593 | . 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸) | 
| 5 | 2 | rnmpt 5967 | . . 3
⊢ ran 𝐺 = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} | 
| 6 |  | fofn 6821 | . . . . . 6
⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → 𝐹 Fn (1...𝑁)) | 
| 7 |  | fnrnfv 6967 | . . . . . 6
⊢ (𝐹 Fn (1...𝑁) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)}) | 
| 8 | 6, 7 | syl 17 | . . . . 5
⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)}) | 
| 9 | 8 | adantl 481 | . . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)}) | 
| 10 |  | df-fo 6566 | . . . . . . 7
⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 ↔ (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸)) | 
| 11 | 10 | biimpi 216 | . . . . . 6
⊢ (𝐹:(1...𝑁)–onto→dom 𝐸 → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸)) | 
| 12 | 11 | adantl 481 | . . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸)) | 
| 13 |  | eqeq1 2740 | . . . . . . . . 9
⊢ (ran
𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ dom 𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)})) | 
| 14 |  | eqcom 2743 | . . . . . . . . 9
⊢ (dom
𝐸 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸) | 
| 15 | 13, 14 | bitrdi 287 | . . . . . . . 8
⊢ (ran
𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} ↔ {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸)) | 
| 16 |  | ffn 6735 | . . . . . . . . . . . . . 14
⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁)) | 
| 17 |  | fseq1hash 14416 | . . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁) | 
| 18 | 16, 17 | sylan2 593 | . . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁) | 
| 19 | 1, 18 | sylan2 593 | . . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (♯‘𝐹) = 𝑁) | 
| 20 |  | fz0add1fz1 13775 | . . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (1...𝑁)) | 
| 21 |  | nn0z 12640 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) | 
| 22 |  | fzval3 13774 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℤ →
(1...𝑁) = (1..^(𝑁 + 1))) | 
| 23 | 21, 22 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ (1...𝑁) =
(1..^(𝑁 +
1))) | 
| 24 |  | nn0cn 12538 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) | 
| 25 |  | 1cnd 11257 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ0
→ 1 ∈ ℂ) | 
| 26 | 24, 25 | addcomd 11464 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) = (1 + 𝑁)) | 
| 27 | 26 | oveq2d 7448 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ (1..^(𝑁 + 1)) =
(1..^(1 + 𝑁))) | 
| 28 | 23, 27 | eqtrd 2776 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ (1...𝑁) = (1..^(1 +
𝑁))) | 
| 29 | 28 | eleq2d 2826 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ0
→ (𝑧 ∈ (1...𝑁) ↔ 𝑧 ∈ (1..^(1 + 𝑁)))) | 
| 30 | 29 | biimpa 476 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ (1..^(1 + 𝑁))) | 
| 31 | 21 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → 𝑁 ∈ ℤ) | 
| 32 |  | fzosubel3 13766 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ (1..^(1 + 𝑁)) ∧ 𝑁 ∈ ℤ) → (𝑧 − 1) ∈ (0..^𝑁)) | 
| 33 | 30, 31, 32 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → (𝑧 − 1) ∈ (0..^𝑁)) | 
| 34 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑧 − 1) → (𝑥 + 1) = ((𝑧 − 1) + 1)) | 
| 35 | 34 | eqeq2d 2747 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑧 − 1) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1))) | 
| 36 | 35 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) ∧ 𝑥 = (𝑧 − 1)) → (𝑧 = (𝑥 + 1) ↔ 𝑧 = ((𝑧 − 1) + 1))) | 
| 37 |  | elfzelz 13565 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℤ) | 
| 38 | 37 | zcnd 12725 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (1...𝑁) → 𝑧 ∈ ℂ) | 
| 39 | 38 | adantl 481 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → 𝑧 ∈ ℂ) | 
| 40 |  | 1cnd 11257 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → 1 ∈
ℂ) | 
| 41 | 39, 40 | npcand 11625 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → ((𝑧 − 1) + 1) = 𝑧) | 
| 42 | 41 | eqcomd 2742 | . . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → 𝑧 = ((𝑧 − 1) + 1)) | 
| 43 | 33, 36, 42 | rspcedvd 3623 | . . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 ∈ (1...𝑁)) → ∃𝑥 ∈ (0..^𝑁)𝑧 = (𝑥 + 1)) | 
| 44 |  | fveq2 6905 | . . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑥 + 1) → (𝐹‘𝑧) = (𝐹‘(𝑥 + 1))) | 
| 45 | 44 | eqeq2d 2747 | . . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑥 + 1) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 46 | 45 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ0
∧ 𝑧 = (𝑥 + 1)) → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 47 | 20, 43, 46 | rexxfrd 5408 | . . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ (∃𝑧 ∈
(1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 48 | 47 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ (♯‘𝐹) =
𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 49 |  | oveq2 7440 | . . . . . . . . . . . . . . . 16
⊢
((♯‘𝐹) =
𝑁 →
(0..^(♯‘𝐹)) =
(0..^𝑁)) | 
| 50 | 49 | rexeqdv 3326 | . . . . . . . . . . . . . . 15
⊢
((♯‘𝐹) =
𝑁 → (∃𝑥 ∈
(0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 51 | 50 | bibi2d 342 | . . . . . . . . . . . . . 14
⊢
((♯‘𝐹) =
𝑁 → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))) | 
| 52 | 51 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ (♯‘𝐹) =
𝑁) → ((∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))) ↔ (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^𝑁)𝑦 = (𝐹‘(𝑥 + 1))))) | 
| 53 | 48, 52 | mpbird 257 | . . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ (♯‘𝐹) =
𝑁) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 54 | 19, 53 | syldan 591 | . . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1)))) | 
| 55 | 54 | abbidv 2807 | . . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))}) | 
| 56 | 55 | eqeq1d 2738 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 ↔ {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)) | 
| 57 | 56 | biimpcd 249 | . . . . . . . 8
⊢ ({𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)) | 
| 58 | 15, 57 | biimtrdi 253 | . . . . . . 7
⊢ (ran
𝐹 = dom 𝐸 → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))) | 
| 59 | 58 | com23 86 | . . . . . 6
⊢ (ran
𝐹 = dom 𝐸 → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))) | 
| 60 | 59 | adantl 481 | . . . . 5
⊢ ((𝐹 Fn (1...𝑁) ∧ ran 𝐹 = dom 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸))) | 
| 61 | 12, 60 | mpcom 38 | . . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → (ran 𝐹 = {𝑦 ∣ ∃𝑧 ∈ (1...𝑁)𝑦 = (𝐹‘𝑧)} → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸)) | 
| 62 | 9, 61 | mpd 15 | . . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → {𝑦 ∣ ∃𝑥 ∈ (0..^(♯‘𝐹))𝑦 = (𝐹‘(𝑥 + 1))} = dom 𝐸) | 
| 63 | 5, 62 | eqtrid 2788 | . 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → ran 𝐺 = dom 𝐸) | 
| 64 |  | dffo2 6823 | . 2
⊢ (𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ran 𝐺 = dom 𝐸)) | 
| 65 | 4, 63, 64 | sylanbrc 583 | 1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–onto→dom 𝐸) |