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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 4740 | . 2 ⊢ Rel I | |
2 | relcnv 4989 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 5127 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 5015 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2191 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 3203 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 5200 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 937 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ⊆ wss 3121 I cid 4273 ◡ccnv 4610 ∘ ccom 4615 Rel wrel 4616 Fun wfun 5192 |
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 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-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-un 3125 df-in 3127 df-ss 3134 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-fun 5200 |
This theorem is referenced by: cnvresid 5272 fnresi 5315 fvi 5553 ssdomg 6756 climshft2 11269 |
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