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| Description: Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.) | 
| Ref | Expression | 
|---|---|
| resmptf.a | ⊢ Ⅎ𝑥𝐴 | 
| resmptf.b | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| resmptf | ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | resmpt 6054 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)) | |
| 2 | resmptf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
| 4 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑦𝐶 | |
| 5 | nfcsb1v 3922 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
| 6 | csbeq1a 3912 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
| 7 | 2, 3, 4, 5, 6 | cbvmptf 5250 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) | 
| 8 | 7 | reseq1i 5992 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) | 
| 9 | resmptf.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
| 10 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 11 | 9, 10, 4, 5, 6 | cbvmptf 5250 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) | 
| 12 | 1, 8, 11 | 3eqtr4g 2801 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 Ⅎwnfc 2889 ⦋csb 3898 ⊆ wss 3950 ↦ cmpt 5224 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-mpt 5225 df-xp 5690 df-rel 5691 df-res 5696 | 
| This theorem is referenced by: esumval 34048 esumel 34049 esumsplit 34055 esumss 34074 limsupequzmpt2 45738 liminfequzmpt2 45811 | 
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