Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 |
---|---|
ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
Ref | Expression |
---|---|
strfvss | ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | id 22 | . . . 4 ⊢ (𝑆 ∈ V → 𝑆 ∈ V) | |
3 | 1, 2 | strfvnd 16607 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
4 | fvssunirn 6705 | . . 3 ⊢ (𝑆‘𝑁) ⊆ ∪ ran 𝑆 | |
5 | 3, 4 | eqsstrdi 3931 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
6 | fvprc 6668 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
7 | 0ss 4285 | . . 3 ⊢ ∅ ⊆ ∪ ran 𝑆 | |
8 | 6, 7 | eqsstrdi 3931 | . 2 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
9 | 5, 8 | pm2.61i 185 | 1 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 ∅c0 4211 ∪ cuni 4796 ran crn 5526 ‘cfv 6339 Slot cslot 16587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fv 6347 df-slot 16592 |
This theorem is referenced by: wunstr 16612 prdsval 16833 prdsbas 16835 prdsplusg 16836 prdsmulr 16837 prdsvsca 16838 prdshom 16845 |
Copyright terms: Public domain | W3C validator |