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Mirrors > Home > MPE Home > Th. List > funcnvres | Structured version Visualization version GIF version |
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 5702 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | df-rn 5700 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | |
3 | 1, 2 | eqtri 2763 | . . 3 ⊢ (𝐹 “ 𝐴) = dom ◡(𝐹 ↾ 𝐴) |
4 | 3 | reseq2i 5997 | . 2 ⊢ (◡𝐹 ↾ (𝐹 “ 𝐴)) = (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) |
5 | resss 6022 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
6 | cnvss 5886 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 |
8 | funssres 6612 | . . 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 2788 | 1 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3963 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 |
This theorem is referenced by: cnvresid 6647 funcnvres2 6648 f1orescnv 6864 f1imacnv 6865 sbthlem4 9125 fpwwe2lem5 10673 fpwwe2lem8 10676 hmeores 23795 dvcnvrelem2 26072 dfrelog 26622 efopnlem2 26714 diophrw 42747 |
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