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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 46653. 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 12424 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | fzo0 13588 | . . . . 5 ⊢ (0..^0) = ∅ | |
3 | 2 | eqcomi 2745 | . . . 4 ⊢ ∅ = (0..^0) |
4 | 3 | naryfvalel 46648 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (0-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m ∅)⟶𝑋)) |
5 | 1, 4 | mpan 688 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m ∅)⟶𝑋)) |
6 | mapdm0 8776 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ↑m ∅) = {∅}) | |
7 | 6 | feq2d 6651 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:(𝑋 ↑m ∅)⟶𝑋 ↔ 𝐹:{∅}⟶𝑋)) |
8 | 0ex 5262 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | fsn2 7078 | . . . . 5 ⊢ (𝐹:{∅}⟶𝑋 ↔ ((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉})) |
10 | opeq2 4829 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘∅) → 〈∅, 𝑥〉 = 〈∅, (𝐹‘∅)〉) | |
11 | 10 | sneqd 4596 | . . . . . 6 ⊢ (𝑥 = (𝐹‘∅) → {〈∅, 𝑥〉} = {〈∅, (𝐹‘∅)〉}) |
12 | 11 | rspceeqv 3593 | . . . . 5 ⊢ (((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉}) → ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉}) |
13 | 9, 12 | sylbi 216 | . . . 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 6824 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {〈∅, 𝑥〉}:{∅}⟶𝑋) |
17 | feq1 6646 | . . . . . 6 ⊢ (𝐹 = {〈∅, 𝑥〉} → (𝐹:{∅}⟶𝑋 ↔ {〈∅, 𝑥〉}:{∅}⟶𝑋)) | |
18 | 16, 17 | syl5ibrcom 246 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋)) |
19 | 18 | rexlimiv 3143 | . . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋) |
20 | 13, 19 | impbii 208 | . . 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 304 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 Vcvv 3443 ∅c0 4280 {csn 4584 〈cop 4590 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 ↑m cmap 8761 0cc0 11047 ℕ0cn0 12409 ..^cfzo 13559 -aryF cnaryf 46644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-naryf 46645 |
This theorem is referenced by: 0aryfvalelfv 46653 |
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