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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 5946 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
| 2 | simp2 1000 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐹 ∈ 𝑊) | |
| 3 | opexg 4271 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) | |
| 4 | 3 | 3adant2 1018 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
| 5 | fvexg 5594 | . . 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 2291 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ w3a 980 ∈ wcel 2175 Vcvv 2771 〈cop 3635 ‘cfv 5270 (class class class)co 5943 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-cnv 4682 df-dm 4684 df-rn 4685 df-iota 5231 df-fv 5278 df-ov 5946 |
| This theorem is referenced by: mapxpen 6944 seq1g 10606 seqp1g 10609 seqclg 10615 seqm1g 10617 seqfeq4g 10674 prdsplusgfval 13087 prdsmulrfval 13089 imasex 13108 imasival 13109 imasbas 13110 imasplusg 13111 imasmulr 13112 imasaddfnlemg 13117 imasaddvallemg 13118 plusfvalg 13166 plusffng 13168 gsumsplit1r 13201 gsumprval 13202 gsumfzz 13298 gsumwsubmcl 13299 gsumfzcl 13302 grpsubval 13349 mulgval 13429 mulgfng 13431 mulgnngsum 13434 mulg1 13436 mulgnnp1 13437 mulgnndir 13458 subgintm 13505 subrngintm 13945 scafvalg 14040 scaffng 14042 rmodislmodlem 14083 rmodislmod 14084 lsssn0 14103 lss1d 14116 lssintclm 14117 ellspsn 14150 crngridl 14263 metrest 14949 |
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