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Mirrors > Home > MPE Home > Th. List > strfvss | Structured version Visualization version GIF version |
Description: A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
strfvss.e | ⊢ 𝐸 = Slot 𝑁 |
Ref | Expression |
---|---|
strfvss | ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfvss.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | id 22 | . . . 4 ⊢ (𝑆 ∈ V → 𝑆 ∈ V) | |
3 | 1, 2 | strfvnd 16886 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
4 | fvssunirn 6803 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
5 | 3, 4 | eqsstrdi 3975 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
6 | fvprc 6766 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
7 | 0ss 4330 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
8 | 6, 7 | eqsstrdi 3975 | . 2 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
9 | 5, 8 | pm2.61i 182 | 1 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 ∪ cuni 4839 ran crn 5590 ‘cfv 6433 Slot cslot 16882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fv 6441 df-slot 16883 |
This theorem is referenced by: wunstr 16889 prdsvallem 17165 prdsval 17166 prdsbas 17168 prdsplusg 17169 prdsmulr 17170 prdsvsca 17171 prdshom 17178 |
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