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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 4925 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}) | |
2 | df-mpt 4039 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
3 | 2 | reseq1i 4874 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) |
4 | df-mpt 4039 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)} | |
5 | 1, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 {copab 4036 ↦ cmpt 4037 ↾ cres 4600 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-mpt 4039 df-xp 4604 df-rel 4605 df-res 4610 |
This theorem is referenced by: resmpt3 4927 resmptf 4928 resmptd 4929 f1stres 6119 f2ndres 6120 tposss 6205 dftpos2 6220 dftpos4 6222 djuf1olemr 7010 fisumss 11319 isumclim3 11350 expcnv 11431 fprodssdc 11517 tgrest 12710 cnmptid 12822 dvidlemap 13201 dvcnp2cntop 13204 dvmulxxbr 13207 dvcoapbr 13212 dvrecap 13218 |
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