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Mirrors > Home > ILE Home > Th. List > strslss | GIF version |
Description: Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.) |
Ref | Expression |
---|---|
strss.t | ⊢ 𝑇 ∈ V |
strss.f | ⊢ Fun 𝑇 |
strss.s | ⊢ 𝑆 ⊆ 𝑇 |
strslss.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
strss.n | ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 |
Ref | Expression |
---|---|
strslss | ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strslss.e | . . 3 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | strss.t | . . . 4 ⊢ 𝑇 ∈ V | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → 𝑇 ∈ V) |
4 | strss.f | . . . 4 ⊢ Fun 𝑇 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Fun 𝑇) |
6 | strss.s | . . . 4 ⊢ 𝑆 ⊆ 𝑇 | |
7 | 6 | a1i 9 | . . 3 ⊢ (⊤ → 𝑆 ⊆ 𝑇) |
8 | strss.n | . . . 4 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 | |
9 | 8 | a1i 9 | . . 3 ⊢ (⊤ → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
10 | 1, 3, 5, 7, 9 | strslssd 11994 | . 2 ⊢ (⊤ → (𝐸‘𝑇) = (𝐸‘𝑆)) |
11 | 10 | mptru 1340 | 1 ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 〈cop 3525 Fun wfun 5112 ‘cfv 5118 ℕcn 8713 ndxcnx 11945 Slot cslot 11947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-slot 11952 |
This theorem is referenced by: (None) |
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