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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaiinfv | Structured version Visualization version GIF version |
Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
imaiinfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnssres 6463 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) | |
2 | fniinfv 6735 | . . 3 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ ran (𝐹 ↾ 𝐵)) |
4 | fvres 6682 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) | |
5 | 4 | iineq2i 4932 | . . 3 ⊢ ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) |
6 | 5 | eqcomi 2827 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) |
7 | df-ima 5561 | . . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) | |
8 | 7 | inteqi 4871 | . 2 ⊢ ∩ (𝐹 “ 𝐵) = ∩ ran (𝐹 ↾ 𝐵) |
9 | 3, 6, 8 | 3eqtr4g 2878 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ⊆ wss 3933 ∩ cint 4867 ∩ ciin 4911 ran crn 5549 ↾ cres 5550 “ cima 5551 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: elrfirn 39170 |
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