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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 12377 | . 2 ⊢ (⊤ → (𝐸‘𝑇) = (𝐸‘𝑆)) |
11 | 10 | mptru 1351 | 1 ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 〈cop 3573 Fun wfun 5176 ‘cfv 5182 ℕcn 8848 ndxcnx 12328 Slot cslot 12330 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-slot 12335 |
This theorem is referenced by: (None) |
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