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Mirrors > Home > ILE Home > Th. List > ovexg | GIF version |
Description: Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.) |
Ref | Expression |
---|---|
ovexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5899 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | simp2 1000 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐹 ∈ 𝑊) | |
3 | opexg 4246 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | 3 | 3adant2 1018 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
5 | fvexg 5553 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) | |
6 | 2, 4, 5 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
7 | 1, 6 | eqeltrid 2276 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 ∈ wcel 2160 Vcvv 2752 〈cop 3610 ‘cfv 5235 (class class class)co 5896 |
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-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-cnv 4652 df-dm 4654 df-rn 4655 df-iota 5196 df-fv 5243 df-ov 5899 |
This theorem is referenced by: mapxpen 6876 imasex 12782 imasival 12783 imasbas 12784 imasplusg 12785 imasmulr 12786 imasaddfnlemg 12791 imasaddvallemg 12792 plusfvalg 12839 plusffng 12841 grpsubval 12990 mulgval 13064 mulgfng 13066 mulg1 13069 mulgnnp1 13070 mulgnndir 13091 subgintm 13137 subrngintm 13559 scafvalg 13623 scaffng 13625 rmodislmodlem 13666 rmodislmod 13667 lsssn0 13686 lss1d 13699 lssintclm 13700 lspsnel 13733 crngridl 13844 metrest 14463 |
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