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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 5456 | . . 3 ⊢ ◡ I = I | |
2 | 1 | eqcomi 2619 | . 2 ⊢ I = ◡ I |
3 | funi 5834 | . . 3 ⊢ Fun I | |
4 | funeq 5823 | . . 3 ⊢ ( I = ◡ I → (Fun I ↔ Fun ◡ I )) | |
5 | 3, 4 | mpbii 222 | . 2 ⊢ ( I = ◡ I → Fun ◡ I ) |
6 | funcnvres 5881 | . . 3 ⊢ (Fun ◡ I → ◡( I ↾ 𝐴) = (◡ I ↾ ( I “ 𝐴))) | |
7 | imai 5397 | . . . 4 ⊢ ( I “ 𝐴) = 𝐴 | |
8 | 1, 7 | reseq12i 5315 | . . 3 ⊢ (◡ I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴) |
9 | 6, 8 | syl6eq 2660 | . 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 1475 I cid 4948 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 |
This theorem is referenced by: fcoi1 5991 f1oi 6086 relexpcnv 13623 tsrdir 17061 gicref 17536 ssidcn 20869 idqtop 21319 idhmeo 21386 ltrncnvnid 34431 dihmeetlem1N 35597 dihglblem5apreN 35598 diophrw 36340 cnvrcl0 36951 relexpaddss 37029 |
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