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Mirrors > Home > MPE Home > Th. List > fimacnvinrn | Structured version Visualization version GIF version |
Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
fimacnvinrn | ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inpreima 6250 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
2 | funforn 6035 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) | |
3 | fof 6028 | . . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | |
4 | 2, 3 | sylbi 206 | . . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
5 | fimacnv 6255 | . . . 4 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹) |
7 | 6 | ineq2d 3776 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)) |
8 | cnvresima 5541 | . . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹) | |
9 | resdm2 5542 | . . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹 | |
10 | funrel 5821 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
11 | dfrel2 5502 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
12 | 10, 11 | sylib 207 | . . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
13 | 9, 12 | syl5eq 2656 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
14 | 13 | cnveqd 5220 | . . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹) |
15 | 14 | imaeq1d 5384 | . . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴)) |
16 | 8, 15 | syl5eqr 2658 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴)) |
17 | 1, 7, 16 | 3eqtrrd 2649 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Rel wrel 5043 Fun wfun 5798 ⟶wf 5800 –onto→wfo 5802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 |
This theorem is referenced by: fimacnvinrn2 6257 |
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