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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 4411 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | uniexg 4411 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
3 | ssun2 3281 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
4 | dmrnssfld 4861 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
5 | 3, 4 | sstri 3146 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
6 | ssexg 4115 | . . 3 ⊢ ((ran 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → ran 𝐴 ∈ V) | |
7 | 5, 6 | mpan 421 | . 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 2135 Vcvv 2721 ∪ cun 3109 ⊆ wss 3111 ∪ cuni 3783 dom cdm 4598 ran crn 4599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: rnex 4865 imaexg 4952 xpexr2m 5039 elxp4 5085 elxp5 5086 cnvexg 5135 coexg 5142 fvexg 5499 cofunexg 6071 funrnex 6074 abrexexg 6078 2ndvalg 6103 tposexg 6217 iunon 6243 fopwdom 6793 djuexb 7000 focdmex 10689 shftfvalg 10746 ovshftex 10747 restval 12504 txvalex 12801 txval 12802 blbas 12980 xmettxlem 13056 xmettx 13057 |
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