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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 12622 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 0 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) | |
4 | 2, 3 | fsnd 6892 | . . . 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 13664 | . . . . . 6 ⊢ (0...0) = {0} | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (0...0) = {0}) |
12 | 9, 11 | feq12d 6725 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
13 | fveq1 6906 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃‘0) = ({〈0, 𝑁〉}‘0)) | |
14 | 13 | eqeq1d 2737 | . . . 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 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℤcz 12611 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-fz 13545 |
This theorem is referenced by: is0wlk 30146 is0trl 30152 0pthon1 30157 |
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