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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.) (Proof shortened by AV, 7-Nov-2024.) |
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 1135 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐵 ⊆ 𝐴) | |
3 | sseqin2 4230 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
4 | 2, 3 | sylib 218 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = 𝐵) |
5 | ressbas.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
6 | 5, 1 | ressid2 17277 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
7 | 6 | fveq2d 6910 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘𝑊)) |
8 | 1, 4, 7 | 3eqtr4a 2800 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
9 | 8 | 3expib 1121 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
10 | simp2 1136 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ V) | |
11 | 1 | fvexi 6920 | . . . . . . 7 ⊢ 𝐵 ∈ V |
12 | 11 | inex2 5323 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ V |
13 | baseid 17247 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
14 | 13 | setsid 17241 | . . . . . 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 17278 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
17 | 16 | fveq2d 6910 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) |
18 | 15, 17 | eqtr4d 2777 | . . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
19 | 18 | 3expib 1121 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) |
20 | 9, 19 | pm2.61i 182 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
21 | in0 4400 | . . . . 5 ⊢ (𝐴 ∩ ∅) = ∅ | |
22 | fvprc 6898 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
23 | 1, 22 | eqtrid 2786 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅) |
24 | 23 | ineq2d 4227 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (𝐴 ∩ ∅)) |
25 | 21, 24, 22 | 3eqtr4a 2800 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑊)) |
26 | base0 17249 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
27 | 26 | eqcomi 2743 | . . . . 5 ⊢ (Base‘∅) = ∅ |
28 | reldmress 17275 | . . . . 5 ⊢ Rel dom ↾s | |
29 | 27, 5, 28 | oveqprc 17225 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = (Base‘𝑅)) |
30 | 25, 29 | eqtrd 2774 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
31 | 30 | adantr 480 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
32 | 20, 31 | pm2.61ian 812 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 〈cop 4636 ‘cfv 6562 (class class class)co 7430 sSet csts 17196 ndxcnx 17226 Basecbs 17244 ↾s cress 17273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-1cn 11210 ax-addcl 11212 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 |
This theorem is referenced by: ressbasssg 17281 ressbas2 17282 ressbasssOLD 17284 ressress 17293 rescabs 17882 rescabsOLD 17883 resscatc 18162 idresefmnd 18924 smndex1bas 18931 resscntz 19363 idrespermg 19443 opprsubg 20368 subrngpropd 20584 subrgpropd 20624 sralmod 21211 lidlssbas 21240 lidlbas 21241 resstopn 23209 resstps 23210 ressuss 24286 ressxms 24553 ressms 24554 cphsubrglem 25224 cphsscph 25298 resspos 32940 resstos 32941 xrge0base 32998 xrge00 32999 submomnd 33069 suborng 33324 gsumge0cl 46326 sge0tsms 46335 uzlidlring 48078 dmatALTbas 48246 |
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