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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 5005 | . . 3 ⊢ Rel ◡𝐴 | |
2 | relssdmrn 5148 | . . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) |
4 | df-rn 4636 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
5 | rnexg 4891 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
6 | 4, 5 | eqeltrrid 2265 | . . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V) |
7 | dfdm4 4818 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
8 | dmexg 4890 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) | |
9 | 7, 8 | eqeltrrid 2265 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V) |
10 | xpexg 4739 | . . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V) | |
11 | 6, 9, 10 | syl2anc 411 | . 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V) |
12 | ssexg 4141 | . 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V) | |
13 | 3, 11, 12 | sylancr 414 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 × cxp 4623 ◡ccnv 4624 dom cdm 4625 ran crn 4626 Rel wrel 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-rel 4632 df-cnv 4633 df-dm 4635 df-rn 4636 |
This theorem is referenced by: cnvex 5166 relcnvexb 5167 cofunex2g 6108 cnvf1o 6223 brtpos2 6249 tposexg 6256 cnven 6805 cnvct 6806 fopwdom 6833 relcnvfi 6937 ennnfonelemim 12417 isunitd 13206 pw1nct 14612 |
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