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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 5689 | . . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)) | |
2 | funcnvres2 6628 | . . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 2 | rneqd 5937 | . . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | 1, 3 | eqtr4id 2791 | . 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴)) |
5 | df-rn 5687 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
6 | 5 | ineq2i 4209 | . . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹) |
7 | dmres 6003 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹) | |
8 | dfdm4 5895 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴) | |
9 | 6, 7, 8 | 3eqtr2ri 2767 | . 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹) |
10 | 4, 9 | eqtrdi 2788 | 1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∩ cin 3947 ◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Fun wfun 6537 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 |
This theorem is referenced by: funimass1 6630 funimass2 6631 rescnvimafod 7075 isercolllem2 15614 isercolllem3 15615 isercoll 15616 cncls 22785 preimane 31933 fnpreimac 31934 ffsrn 31992 gsumhashmul 32249 zarcmplem 32930 cvmliftlem15 34358 fcoreslem2 45853 imaelsetpreimafv 46142 |
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