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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. (Contributed by NM, 30-Apr-1998.) |
Ref | Expression |
---|---|
funi | ⊢ Fun I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5171 | . 2 ⊢ Rel I | |
2 | relcnv 5422 | . . . . 5 ⊢ Rel ◡ I | |
3 | coi2 5569 | . . . . 5 ⊢ (Rel ◡ I → ( I ∘ ◡ I ) = ◡ I ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ∘ ◡ I ) = ◡ I |
5 | cnvi 5456 | . . . 4 ⊢ ◡ I = I | |
6 | 4, 5 | eqtri 2632 | . . 3 ⊢ ( I ∘ ◡ I ) = I |
7 | 6 | eqimssi 3622 | . 2 ⊢ ( I ∘ ◡ I ) ⊆ I |
8 | df-fun 5806 | . 2 ⊢ (Fun I ↔ (Rel I ∧ ( I ∘ ◡ I ) ⊆ I )) | |
9 | 1, 7, 8 | mpbir2an 957 | 1 ⊢ Fun I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊆ wss 3540 I cid 4948 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: cnvresid 5882 fnresi 5922 fvi 6165 resiexd 6385 ssdomg 7887 tendo02 35093 residfi 40340 |
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