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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 12564 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 0 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) | |
4 | 2, 3 | fsnd 6872 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) |
5 | fvsng 7172 | . . . . 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 513 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) |
8 | 7 | adantr 482 | . 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 13596 | . . . . . 6 ⊢ (0...0) = {0} | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (0...0) = {0}) |
12 | 9, 11 | feq12d 6701 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
13 | fveq1 6886 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃‘0) = ({〈0, 𝑁〉}‘0)) | |
14 | 13 | eqeq1d 2735 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → ((𝑃‘0) = 𝑁 ↔ ({〈0, 𝑁〉}‘0) = 𝑁)) |
15 | 12, 14 | anbi12d 632 | . . 3 ⊢ (𝑃 = {〈0, 𝑁〉} → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) ↔ ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁))) |
16 | 15 | adantl 483 | . 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 257 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4626 〈cop 4632 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 0cc0 11105 ℤcz 12553 ...cfz 13479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-addrcl 11166 ax-rnegex 11176 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7969 df-2nd 7970 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-neg 11442 df-z 12554 df-uz 12818 df-fz 13480 |
This theorem is referenced by: is0wlk 29349 is0trl 29355 0pthon1 29360 |
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