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Mirrors > Home > MPE Home > Th. List > ressbas | Structured version Visualization version GIF version |
Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
Ref | Expression |
---|---|
ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressbas | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
2 | simp1 1054 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐵 ⊆ 𝐴) | |
3 | sseqin2 3779 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
4 | 2, 3 | sylib 207 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = 𝐵) |
5 | ressbas.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
6 | 5, 1 | ressid2 15755 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
7 | 6 | fveq2d 6107 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘𝑊)) |
8 | 1, 4, 7 | 3eqtr4a 2670 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
9 | 8 | 3expib 1260 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
10 | simp2 1055 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ V) | |
11 | fvex 6113 | . . . . . . . 8 ⊢ (Base‘𝑊) ∈ V | |
12 | 1, 11 | eqeltri 2684 | . . . . . . 7 ⊢ 𝐵 ∈ V |
13 | 12 | inex2 4728 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ V |
14 | baseid 15747 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
15 | 14 | setsid 15742 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
16 | 10, 13, 15 | sylancl 693 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
17 | 5, 1 | ressval2 15756 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
18 | 17 | fveq2d 6107 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
19 | 16, 18 | eqtr4d 2647 | . . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
20 | 19 | 3expib 1260 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
21 | 9, 20 | pm2.61i 175 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
22 | 0fv 6137 | . . . . 5 ⊢ (∅‘(Base‘ndx)) = ∅ | |
23 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
24 | 23, 14 | strfvn 15712 | . . . . 5 ⊢ (Base‘∅) = (∅‘(Base‘ndx)) |
25 | in0 3920 | . . . . 5 ⊢ (𝐴 ∩ ∅) = ∅ | |
26 | 22, 24, 25 | 3eqtr4ri 2643 | . . . 4 ⊢ (𝐴 ∩ ∅) = (Base‘∅) |
27 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
28 | 1, 27 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅) |
29 | 28 | ineq2d 3776 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (𝐴 ∩ ∅)) |
30 | reldmress 15753 | . . . . . . 7 ⊢ Rel dom ↾s | |
31 | 30 | ovprc1 6582 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
32 | 5, 31 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
33 | 32 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑅) = (Base‘∅)) |
34 | 26, 29, 33 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
35 | 34 | adantr 480 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
36 | 21, 35 | pm2.61ian 827 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 Basecbs 15695 ↾s cress 15696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 |
This theorem is referenced by: ressbas2 15758 ressbasss 15759 ressress 15765 rescabs 16316 resscatc 16578 resscntz 17587 idrespermg 17654 opprsubg 18459 subrgpropd 18637 sralmod 19008 resstopn 20800 resstps 20801 ressuss 21877 ressxms 22140 ressms 22141 cphsubrglem 22785 resspos 28990 resstos 28991 xrge0base 29016 xrge00 29017 submomnd 29041 suborng 29146 gsumge0cl 39264 sge0tsms 39273 lidlssbas 41712 lidlbas 41713 uzlidlring 41719 dmatALTbas 41984 |
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