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| Mirrors > Home > MPE Home > Th. List > cnvresid | Structured version Visualization version 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 6161 | . . 3 ⊢ ◡ I = I | |
| 2 | 1 | eqcomi 2746 | . 2 ⊢ I = ◡ I |
| 3 | funi 6598 | . . 3 ⊢ Fun I | |
| 4 | funeq 6586 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
| 5 | 3, 4 | mpbii 233 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
| 6 | funcnvres 6644 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
| 7 | imai 6092 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
| 8 | 1, 7 | reseq12i 5995 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
| 9 | 6, 8 | eqtrdi 2793 | . 2 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = ( I ↾ 𝐴)) |
| 10 | 2, 5, 9 | mp2b 10 | 1 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 I cid 5577 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 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-12 2177 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-mo 2540 df-eu 2569 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-in 3958 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-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: fcoi1 6782 f1oi 6886 relexpcnv 15074 tsrdir 18649 gicref 19290 ssidcn 23263 idqtop 23714 idhmeo 23781 bj-iminvid 37196 ltrncnvnid 40129 dihmeetlem1N 41292 dihglblem5apreN 41293 diophrw 42770 cnvrcl0 43638 relexpaddss 43731 |
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