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Mirrors > Home > MPE Home > Th. List > funimacnv | Structured version Visualization version GIF version |
Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.) |
Ref | Expression |
---|---|
funimacnv | ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5532 | . . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)) | |
2 | funcnvres2 6404 | . . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 2 | rneqd 5772 | . . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | 1, 3 | eqtr4id 2852 | . 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴)) |
5 | df-rn 5530 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
6 | 5 | ineq2i 4136 | . . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹) |
7 | dmres 5840 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹) | |
8 | dfdm4 5728 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴) | |
9 | 6, 7, 8 | 3eqtr2ri 2828 | . 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹) |
10 | 4, 9 | eqtrdi 2849 | 1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∩ cin 3880 ◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 Fun wfun 6318 |
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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 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-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-fun 6326 |
This theorem is referenced by: funimass1 6406 funimass2 6407 isercolllem2 15014 isercolllem3 15015 isercoll 15016 cncls 21879 preimane 30433 fnpreimac 30434 ffsrn 30491 gsumhashmul 30741 zarcmplem 31234 cvmliftlem15 32658 imaelsetpreimafv 43912 |
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