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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 6591 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
2 | funforn 6360 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹) | |
3 | fof 6353 | . . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) | |
4 | 2, 3 | sylbi 209 | . . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹) |
5 | fimacnv 6596 | . . . 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 4041 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)) |
8 | cnvresima 5864 | . . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹) | |
9 | resdm2 5865 | . . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹 | |
10 | funrel 6140 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
11 | dfrel2 5824 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
12 | 10, 11 | sylib 210 | . . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
13 | 9, 12 | syl5eq 2873 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹) |
14 | 13 | cnveqd 5530 | . . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹) |
15 | 14 | imaeq1d 5706 | . . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴)) |
16 | 8, 15 | syl5eqr 2875 | . 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴)) |
17 | 1, 7, 16 | 3eqtrrd 2866 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∩ cin 3797 ◡ccnv 5341 dom cdm 5342 ran crn 5343 ↾ cres 5344 “ cima 5345 Rel wrel 5347 Fun wfun 6117 ⟶wf 6119 –onto→wfo 6121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fo 6129 df-fv 6131 |
This theorem is referenced by: fimacnvinrn2 6598 |
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