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Mirrors > Home > ILE Home > Th. List > slotex | GIF version |
Description: Existence of slot value. A corollary of slotslfn 11685. (Contributed by Jim Kingdon, 12-Feb-2023.) |
Ref | Expression |
---|---|
slotslfn.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
Ref | Expression |
---|---|
slotex | ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slotslfn.e | . . 3 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | 1 | slotslfn 11685 | . 2 ⊢ 𝐸 Fn V |
3 | elex 2644 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | funfvex 5357 | . . 3 ⊢ ((Fun 𝐸 ∧ 𝐴 ∈ dom 𝐸) → (𝐸‘𝐴) ∈ V) | |
5 | 4 | funfni 5148 | . 2 ⊢ ((𝐸 Fn V ∧ 𝐴 ∈ V) → (𝐸‘𝐴) ∈ V) |
6 | 2, 3, 5 | sylancr 406 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 Fn wfn 5044 ‘cfv 5049 ℕcn 8520 ndxcnx 11656 Slot cslot 11658 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-iota 5014 df-fun 5051 df-fn 5052 df-fv 5057 df-slot 11663 |
This theorem is referenced by: topnfn 11825 topnvalg 11832 topnidg 11833 |
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