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Mirrors > Home > ILE Home > Th. List > cnvexg | GIF version |
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Ref | Expression |
---|---|
cnvexg | ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4703 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 4841 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4356 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | rnexg 4597 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
6 | 4, 5 | syl5eqelr 2125 | . . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V) |
7 | dfdm4 4527 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
8 | dmexg 4596 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
9 | 7, 8 | syl5eqelr 2125 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V) |
10 | xpexg 4452 | . . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V) | |
11 | 6, 9, 10 | syl2anc 391 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V) |
12 | ssexg 3896 | . 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V) | |
13 | 3, 11, 12 | sylancr 393 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 × cxp 4343 ◡ccnv 4344 dom cdm 4345 ran crn 4346 Rel wrel 4350 |
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-ral 2311 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-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: cnvex 4856 relcnvexb 4857 cofunex2g 5739 cnvf1o 5846 brtpos2 5866 tposexg 5873 cnven 6288 fopwdom 6310 |
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