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Mirrors > Home > ILE Home > Th. List > resmpt | GIF version |
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.) |
Ref | Expression |
---|---|
resmpt | ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab2 4861 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}) | |
2 | df-mpt 3986 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
3 | 2 | reseq1i 4810 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) |
4 | df-mpt 3986 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)} | |
5 | 1, 3, 4 | 3eqtr4g 2195 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 {copab 3983 ↦ cmpt 3984 ↾ cres 4536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-mpt 3986 df-xp 4540 df-rel 4541 df-res 4546 |
This theorem is referenced by: resmpt3 4863 resmptf 4864 resmptd 4865 f1stres 6050 f2ndres 6051 tposss 6136 dftpos2 6151 dftpos4 6153 djuf1olemr 6932 fisumss 11154 isumclim3 11185 expcnv 11266 tgrest 12327 cnmptid 12439 dvidlemap 12818 dvcnp2cntop 12821 dvmulxxbr 12824 dvcoapbr 12829 dvrecap 12835 |
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