Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1fv | Structured version Visualization version GIF version |
Description: A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Proof shortened by AV, 18-Apr-2021.) |
Ref | Expression |
---|---|
1fv | ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 0 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) | |
4 | 2, 3 | fsnd 6632 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) |
5 | fvsng 6919 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → ({〈0, 𝑁〉}‘0) = 𝑁) | |
6 | 1, 5 | mpan 689 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}‘0) = 𝑁) |
7 | 4, 6 | jca 515 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) |
8 | 7 | adantr 484 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) |
9 | id 22 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → 𝑃 = {〈0, 𝑁〉}) | |
10 | fz0sn 13002 | . . . . . 6 ⊢ (0...0) = {0} | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (0...0) = {0}) |
12 | 9, 11 | feq12d 6475 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
13 | fveq1 6644 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃‘0) = ({〈0, 𝑁〉}‘0)) | |
14 | 13 | eqeq1d 2800 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → ((𝑃‘0) = 𝑁 ↔ ({〈0, 𝑁〉}‘0) = 𝑁)) |
15 | 12, 14 | anbi12d 633 | . . 3 ⊢ (𝑃 = {〈0, 𝑁〉} → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) ↔ ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁))) |
16 | 15 | adantl 485 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) ↔ ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁))) |
17 | 8, 16 | mpbird 260 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℤcz 11969 ...cfz 12885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-fz 12886 |
This theorem is referenced by: is0wlk 27902 is0trl 27908 0pthon1 27913 |
Copyright terms: Public domain | W3C validator |