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| Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6696. (Contributed by NM, 30-Apr-1998.) | 
| Ref | Expression | 
|---|---|
| funi | ⊢ Fun I | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reli 5836 | . 2 ⊢ Rel I | |
| 2 | relcnv 6122 | . . . . 5 ⊢ Rel ◡ I | |
| 3 | coi2 6283 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I | 
| 5 | cnvi 6161 | . . . 4 ⊢ ◡ I = I | |
| 6 | 4, 5 | eqtri 2765 | . . 3 ⊢ ( I ∘ ◡ I ) = I | 
| 7 | 6 | eqimssi 4044 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I | 
| 8 | df-fun 6563 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
| 9 | 1, 7, 8 | mpbir2an 711 | 1 ⊢ Fun I | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 ∘ ccom 5689 Rel wrel 5690 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 | 
| This theorem is referenced by: cnvresid 6645 idfn 6696 fvi 6985 resiexd 7236 ssdomg 9040 residfi 9378 bj-funidres 37152 tendo02 40789 grimidvtxedg 47876 | 
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