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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 17159 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
4 | fvssunirn 6933 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
5 | 3, 4 | eqsstrdi 4034 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
6 | fvprc 6892 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
7 | 0ss 4398 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
8 | 6, 7 | eqsstrdi 4034 | . 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 1533 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 ∅c0 4324 ∪ cuni 4910 ran crn 5681 ‘cfv 6551 Slot cslot 17155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fv 6559 df-slot 17156 |
This theorem is referenced by: wunstr 17162 prdsvallem 17441 prdsval 17442 prdsbas 17444 prdsplusg 17445 prdsmulr 17446 prdsvsca 17447 prdshom 17454 |
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