Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > strfvssn | 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.) (Revised by Jim Kingdon, 19-Jan-2023.) |
Ref | Expression |
---|---|
strfvssn.c | ⊢ 𝐸 = Slot 𝑁 |
strfvssn.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strfvssn.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
strfvssn | ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfvssn.c | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
2 | strfvssn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | strfvssn.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | 1, 2, 3 | strnfvnd 12018 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
5 | 3 | elexd 2702 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
6 | fvssunirng 5444 | . . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆) | |
7 | 5, 6 | syl 14 | . 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆) |
8 | 4, 7 | eqsstrd 3138 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 ∪ cuni 3744 ran crn 4548 ‘cfv 5131 ℕcn 8744 Slot cslot 11997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-slot 12002 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |