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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 1133 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐵 ⊆ 𝐴) | |
3 | sseqin2 4142 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
4 | 2, 3 | sylib 221 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = 𝐵) |
5 | ressbas.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
6 | 5, 1 | ressid2 16544 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
7 | 6 | fveq2d 6649 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘𝑊)) |
8 | 1, 4, 7 | 3eqtr4a 2859 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
9 | 8 | 3expib 1119 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
10 | simp2 1134 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ V) | |
11 | 1 | fvexi 6659 | . . . . . . 7 ⊢ 𝐵 ∈ V |
12 | 11 | inex2 5186 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ V |
13 | baseid 16535 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
14 | 13 | setsid 16530 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
15 | 10, 12, 14 | sylancl 589 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
16 | 5, 1 | ressval2 16545 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
17 | 16 | fveq2d 6649 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
18 | 15, 17 | eqtr4d 2836 | . . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
19 | 18 | 3expib 1119 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
20 | 9, 19 | pm2.61i 185 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
21 | 0fv 6684 | . . . . 5 ⊢ (∅‘(Base‘ndx)) = ∅ | |
22 | 0ex 5175 | . . . . . 6 ⊢ ∅ ∈ V | |
23 | 22, 13 | strfvn 16497 | . . . . 5 ⊢ (Base‘∅) = (∅‘(Base‘ndx)) |
24 | in0 4299 | . . . . 5 ⊢ (𝐴 ∩ ∅) = ∅ | |
25 | 21, 23, 24 | 3eqtr4ri 2832 | . . . 4 ⊢ (𝐴 ∩ ∅) = (Base‘∅) |
26 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
27 | 1, 26 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅) |
28 | 27 | ineq2d 4139 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (𝐴 ∩ ∅)) |
29 | reldmress 16542 | . . . . . . 7 ⊢ Rel dom ↾s | |
30 | 29 | ovprc1 7174 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
31 | 5, 30 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
32 | 31 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑅) = (Base‘∅)) |
33 | 25, 28, 32 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
34 | 33 | adantr 484 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
35 | 20, 34 | pm2.61ian 811 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 Basecbs 16475 ↾s cress 16476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 |
This theorem is referenced by: ressbas2 16547 ressbasss 16548 ressress 16554 rescabs 17095 resscatc 17357 idresefmnd 18056 smndex1bas 18063 resscntz 18454 idrespermg 18531 opprsubg 19382 subrgpropd 19563 sralmod 19952 resstopn 21791 resstps 21792 ressuss 22869 ressxms 23132 ressms 23133 cphsubrglem 23782 cphsscph 23855 resspos 30672 resstos 30673 xrge0base 30719 xrge00 30720 submomnd 30761 suborng 30939 gsumge0cl 43010 sge0tsms 43019 lidlssbas 44546 lidlbas 44547 uzlidlring 44553 dmatALTbas 44810 |
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