Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > resmptd | GIF version |
Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
resmptd.b | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
resmptd | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resmptd.b | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
2 | resmpt 4926 | . 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ⊆ wss 3111 ↦ 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: fisumss 11319 fprodssdc 11517 cnmpt1res 12843 bj-charfundcALT 13532 |
Copyright terms: Public domain | W3C validator |