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Mirrors > Home > MPE Home > Th. List > resmpt3 | Structured version Visualization version GIF version |
Description: Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.) |
Ref | Expression |
---|---|
resmpt3 | ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resres 5865 | . 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) | |
2 | ssid 3988 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
3 | resmpt 5904 | . . . 4 ⊢ (𝐴 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
5 | 4 | reseq1i 5848 | . 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) |
6 | inss1 4204 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
7 | resmpt 5904 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) |
9 | 1, 5, 8 | 3eqtr3i 2852 | 1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3934 ⊆ wss 3935 ↦ cmpt 5145 ↾ cres 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-mpt 5146 df-xp 5560 df-rel 5561 df-res 5566 |
This theorem is referenced by: mptima 5940 offres 7683 lo1resb 14920 o1resb 14922 measinb2 31482 eulerpartgbij 31630 imassmpt 41535 limsupresicompt 42035 liminfresicompt 42059 |
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