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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 48556. 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 12541 | . . 3 ⊢ 0 ∈ ℕ0 | |
| 2 | fzo0 13723 | . . . . 5 ⊢ (0..^0) = ∅ | |
| 3 | 2 | eqcomi 2746 | . . . 4 ⊢ ∅ = (0..^0) |
| 4 | 3 | naryfvalel 48551 | . . 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 8882 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ↑m ∅) = {∅}) | |
| 7 | 6 | feq2d 6722 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:(𝑋 ↑m ∅)⟶𝑋 ↔ 𝐹:{∅}⟶𝑋)) |
| 8 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 9 | 8 | fsn2 7156 | . . . . 5 ⊢ (𝐹:{∅}⟶𝑋 ↔ ((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉})) |
| 10 | opeq2 4874 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘∅) → 〈∅, 𝑥〉 = 〈∅, (𝐹‘∅)〉) | |
| 11 | 10 | sneqd 4638 | . . . . . 6 ⊢ (𝑥 = (𝐹‘∅) → {〈∅, 𝑥〉} = {〈∅, (𝐹‘∅)〉}) |
| 12 | 11 | rspceeqv 3645 | . . . . 5 ⊢ (((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉}) → ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉}) |
| 13 | 9, 12 | sylbi 217 | . . . 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 6891 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {〈∅, 𝑥〉}:{∅}⟶𝑋) |
| 17 | feq1 6716 | . . . . . 6 ⊢ (𝐹 = {〈∅, 𝑥〉} → (𝐹:{∅}⟶𝑋 ↔ {〈∅, 𝑥〉}:{∅}⟶𝑋)) | |
| 18 | 16, 17 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋)) |
| 19 | 18 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋) |
| 20 | 13, 19 | impbii 209 | . . 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 305 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 ℕ0cn0 12526 ..^cfzo 13694 -aryF cnaryf 48547 |
| 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-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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-reu 3381 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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 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-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-naryf 48548 |
| This theorem is referenced by: 0aryfvalelfv 48556 |
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