Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0map0sn0 | Structured version Visualization version GIF version |
Description: The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.) |
Ref | Expression |
---|---|
0map0sn0 | ⊢ (∅ ↑m ∅) = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0bi 6555 | . . 3 ⊢ (𝑓:∅⟶∅ ↔ 𝑓 = ∅) | |
2 | 1 | abbii 2885 | . 2 ⊢ {𝑓 ∣ 𝑓:∅⟶∅} = {𝑓 ∣ 𝑓 = ∅} |
3 | 0ex 5204 | . . 3 ⊢ ∅ ∈ V | |
4 | 3, 3 | mapval 8411 | . 2 ⊢ (∅ ↑m ∅) = {𝑓 ∣ 𝑓:∅⟶∅} |
5 | df-sn 4561 | . 2 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅} | |
6 | 2, 4, 5 | 3eqtr4i 2853 | 1 ⊢ (∅ ↑m ∅) = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {cab 2798 ∅c0 4284 {csn 4560 ⟶wf 6344 (class class class)co 7149 ↑m cmap 8399 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-map 8401 |
This theorem is referenced by: efmndbas0 18051 symgvalstruct 18520 |
Copyright terms: Public domain | W3C validator |