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Mirrors > Home > ILE Home > Th. List > rnexg | GIF version |
Description: The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) |
Ref | Expression |
---|---|
rnexg | ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4175 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | uniexg 4175 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
3 | ssun2 3107 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
4 | dmrnssfld 4595 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
5 | 3, 4 | sstri 2954 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
6 | ssexg 3896 | . . 3 ⊢ ((ran 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → ran 𝐴 ∈ V) | |
7 | 5, 6 | mpan 400 | . 2 ⊢ (∪ ∪ 𝐴 ∈ V → ran 𝐴 ∈ V) |
8 | 1, 2, 7 | 3syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 Vcvv 2557 ∪ cun 2915 ⊆ wss 2917 ∪ cuni 3580 dom cdm 4345 ran crn 4346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: rnex 4599 imaexg 4680 xpexr2m 4762 elxp4 4808 elxp5 4809 cnvexg 4855 coexg 4862 fvexg 5194 cofunexg 5738 funrnex 5741 abrexexg 5745 2ndvalg 5770 tposexg 5873 iunon 5899 fopwdom 6310 shftfvalg 9419 ovshftex 9420 |
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