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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0aryfvalel | Structured version Visualization version GIF version |
Description: A nullary (endo)function on a set 𝑋 is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: (𝐹‘∅) = 𝐶 with 𝐶 ∈ 𝑋, see also 0aryfvalelfv 45399. Instead of (𝐹‘∅), nullary functions are usually written as 𝐹() in literature. (Contributed by AV, 15-May-2024.) |
Ref | Expression |
---|---|
0aryfvalel | ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11934 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | fzo0 13095 | . . . . 5 ⊢ (0..^0) = ∅ | |
3 | 2 | eqcomi 2768 | . . . 4 ⊢ ∅ = (0..^0) |
4 | 3 | naryfvalel 45394 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (0-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m ∅)⟶𝑋)) |
5 | 1, 4 | mpan 690 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m ∅)⟶𝑋)) |
6 | mapdm0 8424 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ↑m ∅) = {∅}) | |
7 | 6 | feq2d 6477 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:(𝑋 ↑m ∅)⟶𝑋 ↔ 𝐹:{∅}⟶𝑋)) |
8 | 0ex 5170 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | fsn2 6882 | . . . . 5 ⊢ (𝐹:{∅}⟶𝑋 ↔ ((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉})) |
10 | opeq2 4756 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘∅) → 〈∅, 𝑥〉 = 〈∅, (𝐹‘∅)〉) | |
11 | 10 | sneqd 4527 | . . . . . 6 ⊢ (𝑥 = (𝐹‘∅) → {〈∅, 𝑥〉} = {〈∅, (𝐹‘∅)〉}) |
12 | 11 | rspceeqv 3554 | . . . . 5 ⊢ (((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉}) → ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉}) |
13 | 9, 12 | sylbi 220 | . . . 4 ⊢ (𝐹:{∅}⟶𝑋 → ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉}) |
14 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → ∅ ∈ V) |
15 | id 22 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋) | |
16 | 14, 15 | fsnd 6637 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {〈∅, 𝑥〉}:{∅}⟶𝑋) |
17 | feq1 6472 | . . . . . 6 ⊢ (𝐹 = {〈∅, 𝑥〉} → (𝐹:{∅}⟶𝑋 ↔ {〈∅, 𝑥〉}:{∅}⟶𝑋)) | |
18 | 16, 17 | syl5ibrcom 250 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋)) |
19 | 18 | rexlimiv 3202 | . . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋) |
20 | 13, 19 | impbii 212 | . . 3 ⊢ (𝐹:{∅}⟶𝑋 ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉}) |
21 | 20 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:{∅}⟶𝑋 ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
22 | 5, 7, 21 | 3bitrd 309 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 Vcvv 3407 ∅c0 4221 {csn 4515 〈cop 4521 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 ↑m cmap 8409 0cc0 10560 ℕ0cn0 11919 ..^cfzo 13067 -aryF cnaryf 45390 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-naryf 45391 |
This theorem is referenced by: 0aryfvalelfv 45399 |
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