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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 4887 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 5029 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4520 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | rnexg 4774 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
6 | 4, 5 | eqeltrrid 2205 | . . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V) |
7 | dfdm4 4701 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
8 | dmexg 4773 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
9 | 7, 8 | eqeltrrid 2205 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V) |
10 | xpexg 4623 | . . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V) | |
11 | 6, 9, 10 | syl2anc 408 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V) |
12 | ssexg 4037 | . 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V) | |
13 | 3, 11, 12 | sylancr 410 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 × cxp 4507 ◡ccnv 4508 dom cdm 4509 ran crn 4510 Rel wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: cnvex 5047 relcnvexb 5048 cofunex2g 5978 cnvf1o 6090 brtpos2 6116 tposexg 6123 cnven 6670 cnvct 6671 fopwdom 6698 relcnvfi 6797 ennnfonelemim 11864 |
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