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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 4992 | . . 3 ⊢ ◡ I = I | |
2 | 1 | eqcomi 2161 | . 2 ⊢ I = ◡ I |
3 | funi 5204 | . . 3 ⊢ Fun I | |
4 | funeq 5192 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
5 | 3, 4 | mpbii 147 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
6 | funcnvres 5245 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
7 | imai 4944 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
8 | 1, 7 | reseq12i 4866 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
9 | 6, 8 | eqtrdi 2206 | . 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 1335 I cid 4250 ◡ccnv 4587 ↾ cres 4590 “ cima 4591 Fun wfun 5166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-fun 5174 |
This theorem is referenced by: fcoi1 5352 f1oi 5454 ssidcn 12680 idhmeo 12787 |
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