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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 mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.) |
Ref | Expression |
---|---|
fimacnv | ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 5907 | . . 3 ⊢ (◡𝐹 “ 𝐵) ⊆ ran ◡𝐹 | |
2 | dfdm4 5728 | . . . 4 ⊢ dom 𝐹 = ran ◡𝐹 | |
3 | fdm 6495 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | ssid 3937 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
5 | 3, 4 | eqsstrdi 3969 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
6 | 2, 5 | eqsstrrid 3964 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ◡𝐹 ⊆ 𝐴) |
7 | 1, 6 | sstrid 3926 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) ⊆ 𝐴) |
8 | fimass 6529 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵) | |
9 | ffun 6490 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
10 | 4, 3 | sseqtrrid 3968 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
11 | funimass3 6801 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | |
12 | 9, 10, 11 | syl2anc 587 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
13 | 8, 12 | mpbid 235 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (◡𝐹 “ 𝐵)) |
14 | 7, 13 | eqssd 3932 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ⊆ wss 3881 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 Fun wfun 6318 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 |
This theorem is referenced by: fimacnvinrn 6817 fmpt 6851 frnsuppeq 7825 fin1a2lem7 9817 cnclima 21873 iscncl 21874 cnindis 21897 cncmp 21997 ptrescn 22244 qtopuni 22307 qtopcld 22318 qtopcmap 22324 ordthmeolem 22406 rnelfmlem 22557 mbfdm 24230 ismbf 24232 mbfimaicc 24235 ismbf2d 24244 ismbf3d 24258 mbfimaopn2 24261 i1fd 24285 plyeq0 24808 elrspunidl 31014 fsumcvg4 31303 zrhunitpreima 31329 imambfm 31630 carsggect 31686 dstrvprob 31839 poimirlem30 35087 dvtan 35107 smfresal 43420 |
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