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| Description: The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) | 
| Ref | Expression | 
|---|---|
| funcnvres | ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ima 5697 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
| 2 | df-rn 5695 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | |
| 3 | 1, 2 | eqtri 2764 | . . 3 ⊢ (𝐹 “ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | 
| 4 | 3 | reseq2i 5993 | . 2 ⊢ (◡𝐹 ↾ (𝐹 “ 𝐴)) = (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) | 
| 5 | resss 6018 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
| 6 | cnvss 5882 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
| 7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 | 
| 8 | funssres 6609 | . . 3 ⊢ ((Fun ◡𝐹 ∧ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) | |
| 9 | 7, 8 | mpan2 691 | . 2 ⊢ (Fun ◡𝐹 → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) | 
| 10 | 4, 9 | eqtr2id 2789 | 1 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ⊆ wss 3950 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 Fun wfun 6554 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 | 
| This theorem is referenced by: cnvresid 6644 funcnvres2 6645 f1orescnv 6862 f1imacnv 6863 sbthlem4 9127 fpwwe2lem5 10676 fpwwe2lem8 10679 hmeores 23780 dvcnvrelem2 26058 dfrelog 26608 efopnlem2 26700 diophrw 42775 | 
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