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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 4774 | . 2 ⊢ Rel I | |
2 | relcnv 5024 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 5163 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 5051 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2210 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 3226 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 5237 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 944 | 1 ⊢ Fun I |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ⊆ wss 3144 I cid 4306 ◡ccnv 4643 ∘ ccom 4648 Rel wrel 4649 Fun wfun 5229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-fun 5237 |
This theorem is referenced by: cnvresid 5309 fnresi 5352 fvi 5594 ssdomg 6805 climshft2 11349 |
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