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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 5905 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)) | |
2 | resmptf.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑦𝐶 | |
5 | nfcsb1v 3907 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
6 | csbeq1a 3897 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
7 | 2, 3, 4, 5, 6 | cbvmptf 5165 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
8 | 7 | reseq1i 5849 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) |
9 | resmptf.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
10 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
11 | 9, 10, 4, 5, 6 | cbvmptf 5165 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
12 | 1, 8, 11 | 3eqtr4g 2881 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2961 ⦋csb 3883 ⊆ wss 3936 ↦ cmpt 5146 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: esumval 31305 esumel 31306 esumsplit 31312 esumss 31331 limsupequzmpt2 42019 liminfequzmpt2 42092 |
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