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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 17065 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
4 | fvssunirn 6879 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
5 | 3, 4 | eqsstrdi 4002 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
6 | fvprc 6838 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
7 | 0ss 4360 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
8 | 6, 7 | eqsstrdi 4002 | . 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 1542 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 ∪ cuni 4869 ran crn 5638 ‘cfv 6500 Slot cslot 17061 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fv 6508 df-slot 17062 |
This theorem is referenced by: wunstr 17068 prdsvallem 17344 prdsval 17345 prdsbas 17347 prdsplusg 17348 prdsmulr 17349 prdsvsca 17350 prdshom 17357 |
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