Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5570 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | df-rn 5568 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | |
3 | 1, 2 | eqtri 2846 | . . 3 ⊢ (𝐹 “ 𝐴) = dom ◡(𝐹 ↾ 𝐴) |
4 | 3 | reseq2i 5852 | . 2 ⊢ (◡𝐹 ↾ (𝐹 “ 𝐴)) = (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) |
5 | resss 5880 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
6 | cnvss 5745 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 |
8 | funssres 6400 | . . 3 ⊢ ((Fun ◡𝐹 ∧ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) | |
9 | 7, 8 | mpan2 689 | . 2 ⊢ (Fun ◡𝐹 → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) |
10 | 4, 9 | syl5req 2871 | 1 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3938 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ↾ 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: cnvresid 6435 funcnvres2 6436 f1orescnv 6632 f1imacnv 6633 sbthlem4 8632 fpwwe2lem6 10059 fpwwe2lem9 10062 hmeores 22381 dvcnvrelem2 24617 dfrelog 25151 efopnlem2 25242 diophrw 39363 |
Copyright terms: Public domain | W3C validator |