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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 5991 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)) | |
2 | resmptf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑦𝐶 | |
5 | nfcsb1v 3880 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
6 | csbeq1a 3869 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
7 | 2, 3, 4, 5, 6 | cbvmptf 5214 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
8 | 7 | reseq1i 5933 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) |
9 | resmptf.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
10 | nfcv 2907 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
11 | 9, 10, 4, 5, 6 | cbvmptf 5214 | . 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 1541 Ⅎwnfc 2887 ⦋csb 3855 ⊆ wss 3910 ↦ cmpt 5188 ↾ cres 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-opab 5168 df-mpt 5189 df-xp 5639 df-rel 5640 df-res 5645 |
This theorem is referenced by: esumval 32636 esumel 32637 esumsplit 32643 esumss 32662 limsupequzmpt2 43931 liminfequzmpt2 44004 |
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