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| 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 1136 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐵 ⊆ 𝐴) | |
| 3 | sseqin2 4222 | . . . . . 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 17279 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) | 
| 7 | 6 | fveq2d 6909 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘𝑊)) | 
| 8 | 1, 4, 7 | 3eqtr4a 2802 | . . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) | 
| 9 | 8 | 3expib 1122 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅))) | 
| 10 | simp2 1137 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ V) | |
| 11 | 1 | fvexi 6919 | . . . . . . 7 ⊢ 𝐵 ∈ V | 
| 12 | 11 | inex2 5317 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ V | 
| 13 | baseid 17251 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
| 14 | 13 | setsid 17245 | . . . . . 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 17280 | . . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) | 
| 17 | 16 | fveq2d 6909 | . . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (Base‘𝑅) = (Base‘(𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉))) | 
| 18 | 15, 17 | eqtr4d 2779 | . . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ 𝐵) = (Base‘𝑅)) | 
| 19 | 18 | 3expib 1122 | . . 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 4394 | . . . . 5 ⊢ (𝐴 ∩ ∅) = ∅ | |
| 22 | fvprc 6897 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅) | |
| 23 | 1, 22 | eqtrid 2788 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅) | 
| 24 | 23 | ineq2d 4219 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (𝐴 ∩ ∅)) | 
| 25 | 21, 24, 22 | 3eqtr4a 2802 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑊)) | 
| 26 | base0 17253 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
| 27 | 26 | eqcomi 2745 | . . . . 5 ⊢ (Base‘∅) = ∅ | 
| 28 | reldmress 17277 | . . . . 5 ⊢ Rel dom ↾s | |
| 29 | 27, 5, 28 | oveqprc 17230 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = (Base‘𝑅)) | 
| 30 | 25, 29 | eqtrd 2776 | . . 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 811 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 〈cop 4631 ‘cfv 6560 (class class class)co 7432 sSet csts 17201 ndxcnx 17231 Basecbs 17248 ↾s cress 17275 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 | 
| This theorem is referenced by: ressbasssg 17283 ressbas2 17284 ressbasssOLD 17286 ressress 17294 rescabs 17878 rescabsOLD 17879 resscatc 18155 idresefmnd 18913 smndex1bas 18920 resscntz 19352 idrespermg 19430 opprsubg 20353 subrngpropd 20569 subrgpropd 20609 sralmod 21195 lidlssbas 21224 lidlbas 21225 resstopn 23195 resstps 23196 ressuss 24272 ressxms 24539 ressms 24540 cphsubrglem 25212 cphsscph 25286 resspos 32957 resstos 32958 xrge0base 33017 xrge00 33018 submomnd 33088 suborng 33346 gsumge0cl 46391 sge0tsms 46400 uzlidlring 48156 dmatALTbas 48323 | 
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