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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 12624 | . . . . . 6 ⊢ 0 ∈ ℤ | |
| 2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 0 ∈ ℤ) | 
| 3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) | |
| 4 | 2, 3 | fsnd 6891 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) | 
| 5 | fvsng 7200 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → ({〈0, 𝑁〉}‘0) = 𝑁) | |
| 6 | 1, 5 | mpan 690 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}‘0) = 𝑁) | 
| 7 | 4, 6 | jca 511 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}:{0}⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) | 
| 8 | 7 | adantr 480 | . 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 13667 | . . . . . 6 ⊢ (0...0) = {0} | |
| 11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (0...0) = {0}) | 
| 12 | 9, 11 | feq12d 6724 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) | 
| 13 | fveq1 6905 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃‘0) = ({〈0, 𝑁〉}‘0)) | |
| 14 | 13 | eqeq1d 2739 | . . . 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 481 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℤcz 12613 ...cfz 13547 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 | 
| This theorem is referenced by: is0wlk 30136 is0trl 30142 0pthon1 30147 | 
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