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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 1128 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐵 ⊆ 𝐴) | |
3 | sseqin2 4189 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
4 | 2, 3 | sylib 219 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = 𝐵) |
5 | ressbas.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
6 | 5, 1 | ressid2 16540 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
7 | 6 | fveq2d 6667 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘𝑊)) |
8 | 1, 4, 7 | 3eqtr4a 2879 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
9 | 8 | 3expib 1114 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
10 | simp2 1129 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ V) | |
11 | 1 | fvexi 6677 | . . . . . . 7 ⊢ 𝐵 ∈ V |
12 | 11 | inex2 5213 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ V |
13 | baseid 16531 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
14 | 13 | setsid 16526 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
15 | 10, 12, 14 | sylancl 586 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
16 | 5, 1 | ressval2 16541 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
17 | 16 | fveq2d 6667 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
18 | 15, 17 | eqtr4d 2856 | . . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
19 | 18 | 3expib 1114 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
20 | 9, 19 | pm2.61i 183 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
21 | 0fv 6702 | . . . . 5 ⊢ (∅‘(Base‘ndx)) = ∅ | |
22 | 0ex 5202 | . . . . . 6 ⊢ ∅ ∈ V | |
23 | 22, 13 | strfvn 16493 | . . . . 5 ⊢ (Base‘∅) = (∅‘(Base‘ndx)) |
24 | in0 4342 | . . . . 5 ⊢ (𝐴 ∩ ∅) = ∅ | |
25 | 21, 23, 24 | 3eqtr4ri 2852 | . . . 4 ⊢ (𝐴 ∩ ∅) = (Base‘∅) |
26 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
27 | 1, 26 | syl5eq 2865 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅) |
28 | 27 | ineq2d 4186 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (𝐴 ∩ ∅)) |
29 | reldmress 16538 | . . . . . . 7 ⊢ Rel dom ↾s | |
30 | 29 | ovprc1 7184 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
31 | 5, 30 | syl5eq 2865 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
32 | 31 | fveq2d 6667 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑅) = (Base‘∅)) |
33 | 25, 28, 32 | 3eqtr4a 2879 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
34 | 33 | adantr 481 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
35 | 20, 34 | pm2.61ian 808 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 〈cop 4563 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 sSet csts 16469 Basecbs 16471 ↾s cress 16472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 |
This theorem is referenced by: ressbas2 16543 ressbasss 16544 ressress 16550 rescabs 17091 resscatc 17353 resscntz 18400 idrespermg 18468 opprsubg 19315 subrgpropd 19499 sralmod 19888 resstopn 21722 resstps 21723 ressuss 22799 ressxms 23062 ressms 23063 cphsubrglem 23708 cphsscph 23781 resspos 30573 resstos 30574 xrge0base 30599 xrge00 30600 submomnd 30638 suborng 30815 gsumge0cl 42530 sge0tsms 42539 idresefmnd 43999 smndex1bas 44006 lidlssbas 44121 lidlbas 44122 uzlidlring 44128 dmatALTbas 44384 |
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