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Mirrors > Home > ILE Home > Th. List > funi | GIF version |
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
Ref | Expression |
---|---|
funi | ⊢ Fun I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 4663 | . 2 ⊢ Rel I | |
2 | relcnv 4912 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 5050 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 4938 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2158 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 3148 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 5120 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 926 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3066 I cid 4205 ◡ccnv 4533 ∘ ccom 4538 Rel wrel 4539 Fun wfun 5112 |
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-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-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-un 3070 df-in 3072 df-ss 3079 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-fun 5120 |
This theorem is referenced by: cnvresid 5192 fnresi 5235 fvi 5471 ssdomg 6665 climshft2 11068 |
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