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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 5388 | . 2 ⊢ ∅:∅⟶𝐴 | |
2 | fun0 5256 | . . 3 ⊢ Fun ∅ | |
3 | cnv0 5014 | . . . 4 ⊢ ◡∅ = ∅ | |
4 | 3 | funeqi 5219 | . . 3 ⊢ (Fun ◡∅ ↔ Fun ∅) |
5 | 2, 4 | mpbir 145 | . 2 ⊢ Fun ◡∅ |
6 | df-f1 5203 | . 2 ⊢ (∅:∅–1-1→𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ◡∅)) | |
7 | 1, 5, 6 | mpbir2an 937 | 1 ⊢ ∅:∅–1-1→𝐴 |
Colors of variables: wff set class |
Syntax hints: ∅c0 3414 ◡ccnv 4610 Fun wfun 5192 ⟶wf 5194 –1-1→wf1 5195 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 |
This theorem is referenced by: fo00 5478 |
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