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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 6829 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
2 | funforn 6592 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) | |
3 | fof 6585 | . . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | |
4 | 2, 3 | sylbi 219 | . . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
5 | fimacnv 6834 | . . . 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 4189 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)) |
8 | cnvresima 6082 | . . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹) | |
9 | resdm2 6083 | . . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹 | |
10 | funrel 6367 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
11 | dfrel2 6041 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
12 | 10, 11 | sylib 220 | . . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
13 | 9, 12 | syl5eq 2868 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
14 | 13 | cnveqd 5741 | . . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹) |
15 | 14 | imaeq1d 5923 | . . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴)) |
16 | 8, 15 | syl5eqr 2870 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴)) |
17 | 1, 7, 16 | 3eqtrrd 2861 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3935 ◡ccnv 5549 dom cdm 5550 ran crn 5551 ↾ cres 5552 “ cima 5553 Rel wrel 5555 Fun wfun 6344 ⟶wf 6346 –onto→wfo 6348 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fo 6356 df-fv 6358 |
This theorem is referenced by: fimacnvinrn2 6836 |
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