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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 5669 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | simp2 945 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐹 ∈ 𝑊) | |
3 | opexg 4064 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | 3 | 3adant2 963 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
5 | fvexg 5337 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 〈𝐴, 𝐵〉 ∈ V) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) | |
6 | 2, 4, 5 | syl2anc 404 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐹‘〈𝐴, 𝐵〉) ∈ V) |
7 | 1, 6 | syl5eqel 2175 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 925 ∈ wcel 1439 Vcvv 2620 〈cop 3453 ‘cfv 5028 (class class class)co 5666 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-cnv 4460 df-dm 4462 df-rn 4463 df-iota 4993 df-fv 5036 df-ov 5669 |
This theorem is referenced by: mapxpen 6618 |
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