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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 4465 | . 2 ⊢ Rel I | |
2 | relcnv 4703 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 4837 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 7 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 4728 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2060 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 2999 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 4904 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 849 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ⊆ wss 2917 I cid 4025 ◡ccnv 4344 ∘ ccom 4349 Rel wrel 4350 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-fun 4904 |
This theorem is referenced by: cnvresid 4973 fnresi 5016 fvi 5230 ssdomg 6258 climshft2 9827 |
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