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Mirrors > Home > ILE Home > Th. List > dmexg | GIF version |
Description: The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
dmexg | ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4201 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | uniexg 4201 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
3 | ssun1 3136 | . . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
4 | dmrnssfld 4623 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
5 | 3, 4 | sstri 3009 | . . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
6 | ssexg 3925 | . . 3 ⊢ ((dom 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → dom 𝐴 ∈ V) | |
7 | 5, 6 | mpan 415 | . 2 ⊢ (∪ ∪ 𝐴 ∈ V → dom 𝐴 ∈ V) |
8 | 1, 2, 7 | 3syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2602 ∪ cun 2972 ⊆ wss 2974 ∪ cuni 3609 dom cdm 4371 ran crn 4372 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-rex 2355 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-cnv 4379 df-dm 4381 df-rn 4382 |
This theorem is referenced by: dmex 4626 iprc 4628 exse2 4729 xpexr2m 4792 elxp4 4838 cnvexg 4885 coexg 4892 dmfex 5110 cofunexg 5769 offval3 5792 1stvalg 5800 tposexg 5907 erexb 6197 f1vrnfibi 6453 shftfvalg 9844 |
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