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Mirrors > Home > ILE Home > Th. List > f10 | 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 5308 | . 2 ⊢ ∅:∅⟶𝐴 | |
2 | fun0 5176 | . . 3 ⊢ Fun ∅ | |
3 | cnv0 4937 | . . . 4 ⊢ ◡∅ = ∅ | |
4 | 3 | funeqi 5139 | . . 3 ⊢ (Fun ◡∅ ↔ Fun ∅) |
5 | 2, 4 | mpbir 145 | . 2 ⊢ Fun ◡∅ |
6 | df-f1 5123 | . 2 ⊢ (∅:∅–1-1→𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ◡∅)) | |
7 | 1, 5, 6 | mpbir2an 926 | 1 ⊢ ∅:∅–1-1→𝐴 |
Colors of variables: wff set class |
Syntax hints: ∅c0 3358 ◡ccnv 4533 Fun wfun 5112 ⟶wf 5114 –1-1→wf1 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 |
This theorem is referenced by: fo00 5396 |
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