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Mirrors > Home > ILE Home > Th. List > mapdm0 | GIF version |
Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.) |
Ref | Expression |
---|---|
mapdm0 | ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4015 | . . . . 5 ⊢ ∅ ∈ V | |
2 | elmapg 6509 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐵)) | |
3 | 1, 2 | mpan2 419 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐵)) |
4 | f0bi 5273 | . . . 4 ⊢ (𝑓:∅⟶𝐵 ↔ 𝑓 = ∅) | |
5 | 3, 4 | syl6bb 195 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑𝑚 ∅) ↔ 𝑓 = ∅)) |
6 | vex 2660 | . . . 4 ⊢ 𝑓 ∈ V | |
7 | 6 | elsn 3509 | . . 3 ⊢ (𝑓 ∈ {∅} ↔ 𝑓 = ∅) |
8 | 5, 7 | syl6bbr 197 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑓 ∈ (𝐵 ↑𝑚 ∅) ↔ 𝑓 ∈ {∅})) |
9 | 8 | eqrdv 2113 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 ∅) = {∅}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ∈ wcel 1463 Vcvv 2657 ∅c0 3329 {csn 3493 ⟶wf 5077 (class class class)co 5728 ↑𝑚 cmap 6496 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-map 6498 |
This theorem is referenced by: (None) |
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