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Mirrors > Home > MPE Home > Th. List > 1fv | Structured version Visualization version GIF version |
Description: A one value function. (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 11265 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 0 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) | |
4 | 2, 3 | fsnd 6091 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) |
5 | fvsng 6352 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → ({〈0, 𝑁〉}‘0) = 𝑁) | |
6 | 1, 5 | mpan 702 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}‘0) = 𝑁) |
7 | 4, 6 | jca 553 | . . 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 12308 | . . . . . 6 ⊢ (0...0) = {0} | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (0...0) = {0}) |
12 | 9, 11 | feq12d 5946 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
13 | fveq1 6102 | . . . . 5 ⊢ (𝑃 = {〈0, 𝑁〉} → (𝑃‘0) = ({〈0, 𝑁〉}‘0)) | |
14 | 13 | eqeq1d 2612 | . . . 4 ⊢ (𝑃 = {〈0, 𝑁〉} → ((𝑃‘0) = 𝑁 ↔ ({〈0, 𝑁〉}‘0) = 𝑁)) |
15 | 12, 14 | anbi12d 743 | . . 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 246 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {〈0, 𝑁〉}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℤcz 11254 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: 0pthon1 26110 is01wlk 41285 is0Trl 41291 0pthon1-av 41296 |
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