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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 4723 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 4861 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4374 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | rnexg 4615 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
6 | 4, 5 | syl5eqelr 2166 | . . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V) |
7 | dfdm4 4545 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
8 | dmexg 4614 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
9 | 7, 8 | syl5eqelr 2166 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V) |
10 | xpexg 4470 | . . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V) | |
11 | 6, 9, 10 | syl2anc 403 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V) |
12 | ssexg 3917 | . 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V) | |
13 | 3, 11, 12 | sylancr 405 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 × cxp 4361 ◡ccnv 4362 dom cdm 4363 ran crn 4364 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: cnvex 4876 relcnvexb 4877 cofunex2g 5759 cnvf1o 5866 brtpos2 5889 tposexg 5896 cnven 6311 fopwdom 6333 relcnvfi 6391 |
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