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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 4199 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹)) | |
2 | sseqin2 4195 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹) | |
3 | 2 | biimpi 218 | . . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹) |
4 | 3 | adantl 484 | . . . . 5 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹) |
5 | 4 | ineq2d 4192 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹)) |
6 | 1, 5 | syl5eq 2871 | . . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹)) |
7 | 6 | imaeq2d 5932 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹)) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
8 | fimacnvinrn 6843 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹))) | |
9 | 8 | adantr 483 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ ((𝐴 ∩ 𝐵) ∩ ran 𝐹))) |
10 | fimacnvinrn 6843 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) | |
11 | 10 | adantr 483 | . 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) |
12 | 7, 9, 11 | 3eqtr4rd 2870 | 1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∩ cin 3938 ⊆ wss 3939 ◡ccnv 5557 ran crn 5559 “ cima 5561 Fun wfun 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 |
This theorem is referenced by: eulerpartgbij 31634 orvcval4 31722 preimaioomnf 43004 |
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