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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 5701 | . . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)) | |
2 | funcnvres2 6647 | . . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 2 | rneqd 5951 | . . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | 1, 3 | eqtr4id 2793 | . 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴)) |
5 | df-rn 5699 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
6 | 5 | ineq2i 4224 | . . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹) |
7 | dmres 6031 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹) | |
8 | dfdm4 5908 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴) | |
9 | 6, 7, 8 | 3eqtr2ri 2769 | . 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹) |
10 | 4, 9 | eqtrdi 2790 | 1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∩ cin 3961 ◡ccnv 5687 dom cdm 5688 ran crn 5689 ↾ cres 5690 “ cima 5691 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 |
This theorem is referenced by: funimass1 6649 funimass2 6650 rescnvimafod 7092 isercolllem2 15698 isercolllem3 15699 isercoll 15700 cncls 23297 preimane 32686 fnpreimac 32687 ffsrn 32746 gsumhashmul 33046 zarcmplem 33841 cvmliftlem15 35282 fcoreslem2 47013 imaelsetpreimafv 47319 |
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