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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressbasssg | Structured version Visualization version GIF version |
Description: The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.) |
Ref | Expression |
---|---|
ressbasssg.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
ressbasssg.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressbasssg | ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbasssg.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
2 | ressbasssg.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | ressbas 17126 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
4 | ssid 3970 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∩ 𝐵) | |
5 | 3, 4 | eqsstrrdi 4003 | . 2 ⊢ (𝐴 ∈ V → (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)) |
6 | reldmress 17122 | . . . . . 6 ⊢ Rel dom ↾s | |
7 | 6 | ovprc2 7401 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
8 | 1, 7 | eqtrid 2785 | . . . 4 ⊢ (¬ 𝐴 ∈ V → 𝑅 = ∅) |
9 | 8 | fveq2d 6850 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) = (Base‘∅)) |
10 | base0 17096 | . . . 4 ⊢ ∅ = (Base‘∅) | |
11 | 0ss 4360 | . . . 4 ⊢ ∅ ⊆ (𝐴 ∩ 𝐵) | |
12 | 10, 11 | eqsstrri 3983 | . . 3 ⊢ (Base‘∅) ⊆ (𝐴 ∩ 𝐵) |
13 | 9, 12 | eqsstrdi 4002 | . 2 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)) |
14 | 5, 13 | pm2.61i 182 | 1 ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 ‘cfv 6500 (class class class)co 7361 Basecbs 17091 ↾s cress 17120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-1cn 11117 ax-addcl 11119 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 |
This theorem is referenced by: ressbasss2 40719 |
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