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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 5015 | . . 3 ⊢ ◡ I = I | |
2 | 1 | eqcomi 2174 | . 2 ⊢ I = ◡ I |
3 | funi 5230 | . . 3 ⊢ Fun I | |
4 | funeq 5218 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
5 | 3, 4 | mpbii 147 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
6 | funcnvres 5271 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
7 | imai 4967 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
8 | 1, 7 | reseq12i 4889 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
9 | 6, 8 | eqtrdi 2219 | . 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 1348 I cid 4273 ◡ccnv 4610 ↾ cres 4613 “ cima 4614 Fun wfun 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 |
This theorem is referenced by: fcoi1 5378 f1oi 5480 ssidcn 13004 idhmeo 13111 |
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