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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 5568 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | df-rn 5566 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | |
3 | 1, 2 | eqtri 2844 | . . 3 ⊢ (𝐹 “ 𝐴) = dom ◡(𝐹 ↾ 𝐴) |
4 | 3 | reseq2i 5850 | . 2 ⊢ (◡𝐹 ↾ (𝐹 “ 𝐴)) = (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) |
5 | resss 5878 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
6 | cnvss 5743 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 |
8 | funssres 6398 | . . 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 2869 | 1 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3936 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 Fun wfun 6349 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 |
This theorem is referenced by: cnvresid 6433 funcnvres2 6434 f1orescnv 6630 f1imacnv 6631 sbthlem4 8630 fpwwe2lem6 10057 fpwwe2lem9 10060 hmeores 22379 dvcnvrelem2 24615 dfrelog 25149 efopnlem2 25240 diophrw 39376 |
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