Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > f10 | Structured version Visualization version GIF version |
Description: The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.) |
Ref | Expression |
---|---|
f10 | ⊢ ∅:∅–1-1→𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 5999 | . 2 ⊢ ∅:∅⟶𝐴 | |
2 | fun0 5868 | . . 3 ⊢ Fun ∅ | |
3 | cnv0 5454 | . . . 4 ⊢ ◡∅ = ∅ | |
4 | 3 | funeqi 5824 | . . 3 ⊢ (Fun ◡∅ ↔ Fun ∅) |
5 | 2, 4 | mpbir 220 | . 2 ⊢ Fun ◡∅ |
6 | df-f1 5809 | . 2 ⊢ (∅:∅–1-1→𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ◡∅)) | |
7 | 1, 5, 6 | mpbir2an 957 | 1 ⊢ ∅:∅–1-1→𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∅c0 3874 ◡ccnv 5037 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 |
This theorem is referenced by: f10d 6082 fo00 6084 marypha1lem 8222 hashf1 13098 usgr0 40469 |
Copyright terms: Public domain | W3C validator |