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Mirrors > Home > MPE Home > Th. List > ressabs | Structured version Visualization version GIF version |
Description: Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
ressabs | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5329 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑋) → 𝐵 ∈ V) | |
2 | 1 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V) |
3 | ressress 17294 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) | |
4 | 2, 3 | syldan 591 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))) |
5 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
6 | sseqin2 4231 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
7 | 5, 6 | sylib 218 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∩ 𝐵) = 𝐵) |
8 | 7 | oveq2d 7447 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s 𝐵)) |
9 | 4, 8 | eqtrd 2775 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 (class class class)co 7431 ↾s cress 17274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 |
This theorem is referenced by: rescabs 17883 rescabsOLD 17884 rescabs2 17885 subsubmgm 18736 subsubm 18842 subsubg 19180 subgslw 19649 pgpfaclem1 20116 ablfaclem3 20122 subsubrng 20580 subsubrg 20615 subdrgint 20821 lsslss 20977 xrge0cmn 21444 zringunit 21495 cnmsgngrp 21615 psgninv 21618 zrhpsgnmhm 21620 xrge0gsumle 24869 xrge0tsms 24870 reefgim 26509 xrge0tsmsd 33048 nn0omnd 33353 nn0archi 33355 resssra 33617 fedgmullem1 33657 fedgmullem2 33658 fedgmul 33659 algextdeglem1 33723 algextdeglem4 33726 rrhcn 33960 qqtopn 33974 lnmlsslnm 43070 lmhmlnmsplit 43076 gsumge0cl 46327 sge0tsms 46336 amgmlemALT 49034 |
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