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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 17219 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
4 | fvssunirn 6940 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
5 | 3, 4 | eqsstrdi 4050 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
6 | fvprc 6899 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
7 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
8 | 6, 7 | eqsstrdi 4050 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 Slot cslot 17215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-slot 17216 |
This theorem is referenced by: wunstr 17222 prdsvallem 17501 prdsval 17502 prdsbas 17504 prdsplusg 17505 prdsmulr 17506 prdsvsca 17507 prdshom 17514 |
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