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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 5422 | . 2 ⊢ ∅:∅⟶𝐴 | |
2 | fun0 5290 | . . 3 ⊢ Fun ∅ | |
3 | cnv0 5047 | . . . 4 ⊢ ◡∅ = ∅ | |
4 | 3 | funeqi 5253 | . . 3 ⊢ (Fun ◡∅ ↔ Fun ∅) |
5 | 2, 4 | mpbir 146 | . 2 ⊢ Fun ◡∅ |
6 | df-f1 5237 | . 2 ⊢ (∅:∅–1-1→𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ◡∅)) | |
7 | 1, 5, 6 | mpbir2an 944 | 1 ⊢ ∅:∅–1-1→𝐴 |
Colors of variables: wff set class |
Syntax hints: ∅c0 3437 ◡ccnv 4640 Fun wfun 5226 ⟶wf 5228 –1-1→wf1 5229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 |
This theorem is referenced by: fo00 5513 |
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