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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 17222 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
| 4 | fvssunirn 6939 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
| 5 | 3, 4 | eqsstrdi 4028 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
| 6 | fvprc 6898 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
| 7 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
| 8 | 6, 7 | eqsstrdi 4028 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 Slot cslot 17218 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-slot 17219 |
| This theorem is referenced by: wunstr 17225 prdsvallem 17499 prdsval 17500 prdsbas 17502 prdsplusg 17503 prdsmulr 17504 prdsvsca 17505 prdshom 17512 |
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