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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 12668 | . 2 ⊢ (⊤ → (𝐸‘𝑇) = (𝐸‘𝑆)) |
11 | 10 | mptru 1373 | 1 ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ⊤wtru 1365 ∈ wcel 2164 Vcvv 2760 ⊆ wss 3154 〈cop 3622 Fun wfun 5249 ‘cfv 5255 ℕcn 8984 ndxcnx 12618 Slot cslot 12620 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2987 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-iota 5216 df-fun 5257 df-fv 5263 df-slot 12625 |
This theorem is referenced by: (None) |
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