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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 7084 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
2 | funforn 6828 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) | |
3 | fof 6821 | . . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | |
4 | 2, 3 | sylbi 217 | . . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
5 | fimacnv 6759 | . . . 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 4228 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)) |
8 | cnvresima 6252 | . . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹) | |
9 | resdm2 6253 | . . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹 | |
10 | funrel 6585 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
11 | dfrel2 6211 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
12 | 10, 11 | sylib 218 | . . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
13 | 9, 12 | eqtrid 2787 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
14 | 13 | cnveqd 5889 | . . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹) |
15 | 14 | imaeq1d 6079 | . . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴)) |
16 | 8, 15 | eqtr3id 2789 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴)) |
17 | 1, 7, 16 | 3eqtrrd 2780 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Rel wrel 5694 Fun wfun 6557 ⟶wf 6559 –onto→wfo 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 |
This theorem is referenced by: fimacnvinrn2 7092 preiman0 32725 |
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