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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 12452 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
5 | 3 | elexd 2750 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
6 | fvssunirng 5525 | . . 3 ⊢ (𝑁 ∈ V → (𝑆‘𝑁) ⊆ ∪ ran 𝑆) | |
7 | 5, 6 | syl 14 | . 2 ⊢ (𝜑 → (𝑆‘𝑁) ⊆ ∪ ran 𝑆) |
8 | 4, 7 | eqsstrd 3191 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 ∪ cuni 3807 ran crn 4623 ‘cfv 5211 ℕcn 8895 Slot cslot 12431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-iota 5173 df-fun 5213 df-fv 5219 df-slot 12436 |
This theorem is referenced by: (None) |
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