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Theorem cnvexg 5046
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
Assertion
Ref Expression
cnvexg (𝐴𝑉𝐴 ∈ V)

Proof of Theorem cnvexg
StepHypRef Expression
1 relcnv 4887 . . 3 Rel 𝐴
2 relssdmrn 5029 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4520 . . . 4 ran 𝐴 = dom 𝐴
5 rnexg 4774 . . . 4 (𝐴𝑉 → ran 𝐴 ∈ V)
64, 5eqeltrrid 2205 . . 3 (𝐴𝑉 → dom 𝐴 ∈ V)
7 dfdm4 4701 . . . 4 dom 𝐴 = ran 𝐴
8 dmexg 4773 . . . 4 (𝐴𝑉 → dom 𝐴 ∈ V)
97, 8eqeltrrid 2205 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
10 xpexg 4623 . . 3 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 × ran 𝐴) ∈ V)
116, 9, 10syl2anc 408 . 2 (𝐴𝑉 → (dom 𝐴 × ran 𝐴) ∈ V)
12 ssexg 4037 . 2 ((𝐴 ⊆ (dom 𝐴 × ran 𝐴) ∧ (dom 𝐴 × ran 𝐴) ∈ V) → 𝐴 ∈ V)
133, 11, 12sylancr 410 1 (𝐴𝑉𝐴 ∈ 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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