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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 6002 | . . 3 ⊢ ◡ I = I | |
2 | 1 | eqcomi 2832 | . 2 ⊢ I = ◡ I |
3 | funi 6389 | . . 3 ⊢ Fun I | |
4 | funeq 6377 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
5 | 3, 4 | mpbii 235 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
6 | funcnvres 6434 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
7 | imai 5944 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
8 | 1, 7 | reseq12i 5853 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
9 | 6, 8 | syl6eq 2874 | . 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 1537 I cid 5461 ◡ccnv 5556 ↾ cres 5559 “ cima 5560 Fun wfun 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 |
This theorem is referenced by: fcoi1 6554 f1oi 6654 relexpcnv 14396 tsrdir 17850 gicref 18413 ssidcn 21865 idqtop 22316 idhmeo 22383 ltrncnvnid 37265 dihmeetlem1N 38428 dihglblem5apreN 38429 diophrw 39363 cnvrcl0 39992 relexpaddss 40070 |
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