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Mirrors > Home > MPE Home > Th. List > resmptf | Structured version Visualization version GIF version |
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 6057 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)) | |
2 | resmptf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑦𝐶 | |
5 | nfcsb1v 3933 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
6 | csbeq1a 3922 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
7 | 2, 3, 4, 5, 6 | cbvmptf 5257 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
8 | 7 | reseq1i 5996 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) |
9 | resmptf.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
10 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
11 | 9, 10, 4, 5, 6 | cbvmptf 5257 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
12 | 1, 8, 11 | 3eqtr4g 2800 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2888 ⦋csb 3908 ⊆ wss 3963 ↦ cmpt 5231 ↾ cres 5691 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: esumval 34027 esumel 34028 esumsplit 34034 esumss 34053 limsupequzmpt2 45674 liminfequzmpt2 45747 |
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