![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ressval2 | GIF version |
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressval2 | ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressvalsets 12493 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) | |
2 | ressbas.r | . . 3 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
3 | ressbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
4 | 3 | ineq2i 3333 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)) |
5 | 4 | opeq2i 3780 | . . . 4 ⊢ 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉 = 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉 |
6 | 5 | oveq2i 5879 | . . 3 ⊢ (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉) = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) |
7 | 1, 2, 6 | 3eqtr4g 2235 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
8 | 7 | 3adant1 1015 | 1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∩ cin 3128 ⊆ wss 3129 〈cop 3594 ‘cfv 5211 (class class class)co 5868 ndxcnx 12429 sSet csts 12430 Basecbs 12432 ↾s cress 12433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-inn 8896 df-ndx 12435 df-slot 12436 df-base 12438 df-sets 12439 df-iress 12440 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |