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Mirrors > Home > ILE Home > Th. List > slotex | GIF version |
Description: Existence of slot value. A corollary of slotslfn 12538. (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 12538 | . 2 ⊢ 𝐸 Fn V |
3 | elex 2763 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | funfvex 5551 | . . 3 ⊢ ((Fun 𝐸 ∧ 𝐴 ∈ dom 𝐸) → (𝐸‘𝐴) ∈ V) | |
5 | 4 | funfni 5335 | . 2 ⊢ ((𝐸 Fn V ∧ 𝐴 ∈ V) → (𝐸‘𝐴) ∈ V) |
6 | 2, 3, 5 | sylancr 414 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 Fn wfn 5230 ‘cfv 5235 ℕcn 8949 ndxcnx 12509 Slot cslot 12511 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-slot 12516 |
This theorem is referenced by: topnfn 12749 topnvalg 12756 topnidg 12757 imasex 12782 imasival 12783 imasbas 12784 imasplusg 12785 imasmulr 12786 imasaddfn 12794 imasaddval 12795 imasaddf 12796 imasmulfn 12797 imasmulval 12798 imasmulf 12799 qusaddval 12811 qusaddf 12812 qusmulval 12813 qusmulf 12814 xpsval 12828 ismgm 12833 plusfvalg 12839 plusffng 12841 issgrp 12866 ismnddef 12879 grppropstrg 12964 grpsubval 12990 mulgval 13064 mulgfng 13066 mulg1 13069 mulgnnp1 13070 mulgnndir 13091 subgintm 13137 isnsg 13141 fnmgp 13276 mgpvalg 13277 mgpplusgg 13278 mgpex 13279 mgpbasg 13280 mgpscag 13281 mgptsetg 13282 mgpdsg 13284 mgpress 13285 isrng 13288 issrg 13319 isring 13354 opprvalg 13419 opprmulfvalg 13420 opprex 13423 opprsllem 13424 subrngintm 13559 islmod 13607 scaffvalg 13622 scafvalg 13623 scaffng 13625 rmodislmodlem 13666 rmodislmod 13667 lsssn0 13686 lss1d 13699 lssintclm 13700 lspsnel 13733 sraval 13753 sralemg 13754 srascag 13758 sravscag 13759 sraipg 13760 sraex 13762 crngridl 13844 znbaslemnn 13935 |
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