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Mathbox for Stefan O'Rear |
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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 6628 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) | |
2 | fniinfv 6923 | . . 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 6865 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) | |
5 | 4 | iineq2i 4980 | . . 3 ⊢ ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) |
6 | 5 | eqcomi 2742 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ 𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) |
7 | df-ima 5650 | . . 3 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) | |
8 | 7 | inteqi 4915 | . 2 ⊢ ∩ (𝐹 “ 𝐵) = ∩ ran (𝐹 ↾ 𝐵) |
9 | 3, 6, 8 | 3eqtr4g 2798 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3914 ∩ cint 4911 ∩ ciin 4959 ran crn 5638 ↾ cres 5639 “ cima 5640 Fn wfn 6495 ‘cfv 6500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 |
This theorem is referenced by: elrfirn 41065 |
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