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Mirrors > Home > MPE Home > Th. List > strss | Structured version Visualization version GIF version |
Description: Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.) |
Ref | Expression |
---|---|
strss.t | ⊢ 𝑇 ∈ V |
strss.f | ⊢ Fun 𝑇 |
strss.s | ⊢ 𝑆 ⊆ 𝑇 |
strss.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
strss.n | ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 |
Ref | Expression |
---|---|
strss | ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strss.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | strss.t | . . . 4 ⊢ 𝑇 ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → 𝑇 ∈ V) |
4 | strss.f | . . . 4 ⊢ Fun 𝑇 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Fun 𝑇) |
6 | strss.s | . . . 4 ⊢ 𝑆 ⊆ 𝑇 | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → 𝑆 ⊆ 𝑇) |
8 | strss.n | . . . 4 ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
10 | 1, 3, 5, 7, 9 | strssd 15737 | . 2 ⊢ (⊤ → (𝐸‘𝑇) = (𝐸‘𝑆)) |
11 | 10 | trud 1484 | 1 ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 〈cop 4131 Fun wfun 5798 ‘cfv 5804 ndxcnx 15692 Slot cslot 15694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 |
This theorem is referenced by: grpss 17263 |
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