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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 4361 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | uniexg 4361 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
3 | ssun2 3240 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
4 | dmrnssfld 4802 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
5 | 3, 4 | sstri 3106 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
6 | ssexg 4067 | . . 3 ⊢ ((ran 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → ran 𝐴 ∈ V) | |
7 | 5, 6 | mpan 420 | . 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 1480 Vcvv 2686 ∪ cun 3069 ⊆ wss 3071 ∪ cuni 3736 dom cdm 4539 ran crn 4540 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: rnex 4806 imaexg 4893 xpexr2m 4980 elxp4 5026 elxp5 5027 cnvexg 5076 coexg 5083 fvexg 5440 cofunexg 6009 funrnex 6012 abrexexg 6016 2ndvalg 6041 tposexg 6155 iunon 6181 fopwdom 6730 djuexb 6929 focdmex 10533 shftfvalg 10590 ovshftex 10591 restval 12126 txvalex 12423 txval 12424 blbas 12602 xmettxlem 12678 xmettx 12679 |
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