Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 45608. 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 12088 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | fzo0 13249 | . . . . 5 ⊢ (0..^0) = ∅ | |
3 | 2 | eqcomi 2743 | . . . 4 ⊢ ∅ = (0..^0) |
4 | 3 | naryfvalel 45603 | . . 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 8512 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ↑m ∅) = {∅}) | |
7 | 6 | feq2d 6520 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:(𝑋 ↑m ∅)⟶𝑋 ↔ 𝐹:{∅}⟶𝑋)) |
8 | 0ex 5189 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | fsn2 6940 | . . . . 5 ⊢ (𝐹:{∅}⟶𝑋 ↔ ((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉})) |
10 | opeq2 4775 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘∅) → 〈∅, 𝑥〉 = 〈∅, (𝐹‘∅)〉) | |
11 | 10 | sneqd 4543 | . . . . . 6 ⊢ (𝑥 = (𝐹‘∅) → {〈∅, 𝑥〉} = {〈∅, (𝐹‘∅)〉}) |
12 | 11 | rspceeqv 3545 | . . . . 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 6692 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {〈∅, 𝑥〉}:{∅}⟶𝑋) |
17 | feq1 6515 | . . . . . 6 ⊢ (𝐹 = {〈∅, 𝑥〉} → (𝐹:{∅}⟶𝑋 ↔ {〈∅, 𝑥〉}:{∅}⟶𝑋)) | |
18 | 16, 17 | syl5ibrcom 250 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋)) |
19 | 18 | rexlimiv 3192 | . . . 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 308 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {〈∅, 𝑥〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 Vcvv 3401 ∅c0 4227 {csn 4531 〈cop 4537 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 0cc0 10712 ℕ0cn0 12073 ..^cfzo 13221 -aryF cnaryf 45599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-naryf 45600 |
This theorem is referenced by: 0aryfvalelfv 45608 |
Copyright terms: Public domain | W3C validator |