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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 5849 | . 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) | |
2 | ssid 3909 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
3 | resmpt 5890 | . . . 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 5832 | . 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) |
6 | inss1 4129 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
7 | resmpt 5890 | . . 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 2767 | 1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∩ cin 3852 ⊆ wss 3853 ↦ cmpt 5120 ↾ cres 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-mpt 5121 df-xp 5542 df-rel 5543 df-res 5548 |
This theorem is referenced by: mptima 5926 offres 7734 lo1resb 15090 o1resb 15092 measinb2 31857 eulerpartgbij 32005 imassmpt 42423 limsupresicompt 42915 liminfresicompt 42939 |
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