![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ressex | GIF version |
Description: Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
Ref | Expression |
---|---|
ressex | ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressvalsets 12518 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉)) | |
2 | simpl 109 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑊 ∈ 𝑋) | |
3 | basendxnn 12512 | . . . 4 ⊢ (Base‘ndx) ∈ ℕ | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (Base‘ndx) ∈ ℕ) |
5 | inex1g 4139 | . . . 4 ⊢ (𝐴 ∈ 𝑌 → (𝐴 ∩ (Base‘𝑊)) ∈ V) | |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝐴 ∩ (Base‘𝑊)) ∈ V) |
7 | setsex 12488 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ ∧ (𝐴 ∩ (Base‘𝑊)) ∈ V) → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ V) | |
8 | 2, 4, 6, 7 | syl3anc 1238 | . 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑊))〉) ∈ V) |
9 | 1, 8 | eqeltrd 2254 | 1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 〈cop 3595 ‘cfv 5216 (class class class)co 5874 ℕcn 8917 ndxcnx 12453 sSet csts 12454 Basecbs 12456 ↾s cress 12457 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-inn 8918 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 |
This theorem is referenced by: ressressg 12528 mgpress 13094 rdivmuldivd 13266 invrpropdg 13271 |
Copyright terms: Public domain | W3C validator |