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Mirrors > Home > MPE Home > Th. List > funi | Structured version Visualization version GIF version |
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6626. (Contributed by NM, 30-Apr-1998.) |
Ref | Expression |
---|---|
funi | ⊢ Fun I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5780 | . 2 ⊢ Rel I | |
2 | relcnv 6054 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 6213 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 6092 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2765 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 4000 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 6495 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 709 | 1 ⊢ Fun I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊆ wss 3908 I cid 5528 ◡ccnv 5630 ∘ ccom 5635 Rel wrel 5636 Fun wfun 6487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-fun 6495 |
This theorem is referenced by: cnvresid 6577 idfn 6626 fvi 6914 resiexd 7162 ssdomg 8898 residfi 9235 bj-funidres 35560 tendo02 39188 |
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