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Mirrors > Home > ILE Home > Th. List > rnex | 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, 7-Jul-2008.) |
Ref | Expression |
---|---|
dmex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rnex | ⊢ ran 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | rnexg 4853 | . 2 ⊢ (𝐴 ∈ V → ran 𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 Vcvv 2712 ran crn 4589 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-cnv 4596 df-dm 4598 df-rn 4599 |
This theorem is referenced by: ffoss 5448 abrexex 6067 fo2nd 6108 tfrexlem 6283 ixpsnf1o 6683 bren 6694 xpassen 6777 mapen 6793 ssenen 6798 seqex 10355 hashfacen 10718 shftfval 10732 restfn 12425 tgioo 13016 |
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