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Mirrors > Home > ILE Home > Th. List > cnvresid | GIF version |
Description: Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.) |
Ref | Expression |
---|---|
cnvresid | ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvi 4728 | . . 3 ⊢ ◡ I = I | |
2 | 1 | eqcomi 2044 | . 2 ⊢ I = ◡ I |
3 | funi 4932 | . . 3 ⊢ Fun I | |
4 | funeq 4921 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
5 | 3, 4 | mpbii 136 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
6 | funcnvres 4972 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
7 | imai 4681 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
8 | 1, 7 | reseq12i 4610 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
9 | 6, 8 | syl6eq 2088 | . 2 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = ( I ↾ 𝐴)) |
10 | 2, 5, 9 | mp2b 8 | 1 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 I cid 4025 ◡ccnv 4344 ↾ cres 4347 “ cima 4348 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 |
This theorem is referenced by: fcoi1 5070 f1oi 5164 |
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