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Mirrors > Home > MPE Home > Th. List > fimacnvinrn2 | 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, 17-Feb-2017.) |
Ref | Expression |
---|---|
fimacnvinrn2 | ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4086 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹)) | |
2 | sseqin2 4082 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹) | |
3 | 2 | biimpi 208 | . . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹) |
4 | 3 | adantl 474 | . . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹) |
5 | 4 | ineq2d 4079 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹)) |
6 | 1, 5 | syl5eq 2828 | . . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹)) |
7 | 6 | imaeq2d 5775 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
8 | fimacnvinrn 6671 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹))) | |
9 | 8 | adantr 473 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹))) |
10 | fimacnvinrn 6671 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) | |
11 | 10 | adantr 473 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
12 | 7, 9, 11 | 3eqtr4rd 2827 | 1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∩ cin 3830 ⊆ wss 3831 ◡ccnv 5410 ran crn 5412 “ cima 5414 Fun wfun 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pr 5190 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3419 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-br 4935 df-opab 4997 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-fo 6199 df-fv 6201 |
This theorem is referenced by: eulerpartgbij 31307 orvcval4 31396 preimaioomnf 42463 |
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