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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 48485. 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 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | fzo0 13720 | . . . . 5 ⊢ (0..^0) = ∅ | |
3 | 2 | eqcomi 2744 | . . . 4 ⊢ ∅ = (0..^0) |
4 | 3 | naryfvalel 48480 | . . 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 8881 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ↑m ∅) = {∅}) | |
7 | 6 | feq2d 6723 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹:(𝑋 ↑m ∅)⟶𝑋 ↔ 𝐹:{∅}⟶𝑋)) |
8 | 0ex 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | fsn2 7156 | . . . . 5 ⊢ (𝐹:{∅}⟶𝑋 ↔ ((𝐹‘∅) ∈ 𝑋 ∧ 𝐹 = {〈∅, (𝐹‘∅)〉})) |
10 | opeq2 4879 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘∅) → 〈∅, 𝑥〉 = 〈∅, (𝐹‘∅)〉) | |
11 | 10 | sneqd 4643 | . . . . . 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 6892 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → {〈∅, 𝑥〉}:{∅}⟶𝑋) |
17 | feq1 6717 | . . . . . 6 ⊢ (𝐹 = {〈∅, 𝑥〉} → (𝐹:{∅}⟶𝑋 ↔ {〈∅, 𝑥〉}:{∅}⟶𝑋)) | |
18 | 16, 17 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝐹 = {〈∅, 𝑥〉} → 𝐹:{∅}⟶𝑋)) |
19 | 18 | rexlimiv 3146 | . . . 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 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 ℕ0cn0 12524 ..^cfzo 13691 -aryF cnaryf 48476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-naryf 48477 |
This theorem is referenced by: 0aryfvalelfv 48485 |
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