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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 4408 | . 2 ⊢ Rel I | |
2 | relcnv 4646 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 4780 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 7 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 4671 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2057 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 2993 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 4847 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 848 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ⊆ wss 2911 I cid 4016 ◡ccnv 4287 ∘ ccom 4292 Rel wrel 4293 Fun wfun 4839 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-fun 4847 |
This theorem is referenced by: cnvresid 4916 fnresi 4959 fvi 5173 ssdomg 6194 |
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