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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 4718 | . 2 ⊢ Rel I | |
2 | relcnv 4967 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 5105 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 4993 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2178 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 3184 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 5175 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 927 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ⊆ wss 3102 I cid 4251 ◡ccnv 4588 ∘ ccom 4593 Rel wrel 4594 Fun wfun 5167 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-fun 5175 |
This theorem is referenced by: cnvresid 5247 fnresi 5290 fvi 5528 ssdomg 6726 climshft2 11215 |
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