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| Mirrors > Home > MPE Home > Th. List > fimacnv | Structured version Visualization version GIF version | ||
| Description: The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007.) (Proof shortened by AV, 20-Sep-2024.) |
| Ref | Expression |
|---|---|
| fimacnv | ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frn 6743 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
| 2 | cnvimassrndm 6172 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (◡𝐹 “ 𝐵) = dom 𝐹) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = dom 𝐹) |
| 4 | fdm 6745 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
| 5 | 3, 4 | eqtrd 2777 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fn 6564 df-f 6565 |
| This theorem is referenced by: fco 6760 f1co 6815 fimacnvinrn 7091 fmpt 7130 fsuppeq 8200 fsuppeqg 8201 fin1a2lem7 10446 cnclima 23276 iscncl 23277 cnindis 23300 cncmp 23400 ptrescn 23647 qtopuni 23710 qtopcld 23721 qtopcmap 23727 ordthmeolem 23809 rnelfmlem 23960 mbfdm 25661 ismbf 25663 mbfimaicc 25666 ismbf2d 25675 ismbf3d 25689 mbfimaopn2 25692 i1fd 25716 plyeq0 26250 elrspunidl 33456 fsumcvg4 33949 zrhunitpreima 33977 imambfm 34264 carsggect 34320 dstrvprob 34474 poimirlem30 37657 dvtan 37677 smfresal 46803 cnneiima 48814 |
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