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Theorem cnvexg 4955
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 4797 . . 3 Rel 𝐴
2 relssdmrn 4938 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 7 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4439 . . . 4 ran 𝐴 = dom 𝐴
5 rnexg 4686 . . . 4 (𝐴𝑉 → ran 𝐴 ∈ V)
64, 5syl5eqelr 2175 . . 3 (𝐴𝑉 → dom 𝐴 ∈ V)
7 dfdm4 4616 . . . 4 dom 𝐴 = ran 𝐴
8 dmexg 4685 . . . 4 (𝐴𝑉 → dom 𝐴 ∈ V)
97, 8syl5eqelr 2175 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
10 xpexg 4540 . . 3 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 × ran 𝐴) ∈ V)
116, 9, 10syl2anc 403 . 2 (𝐴𝑉 → (dom 𝐴 × ran 𝐴) ∈ V)
12 ssexg 3970 . 2 ((𝐴 ⊆ (dom 𝐴 × ran 𝐴) ∧ (dom 𝐴 × ran 𝐴) ∈ V) → 𝐴 ∈ V)
133, 11, 12sylancr 405 1 (𝐴𝑉𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  Vcvv 2619  wss 2997   × cxp 4426  ccnv 4427  dom cdm 4428  ran crn 4429  Rel wrel 4433
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  cnvex  4956  relcnvexb  4957  cofunex2g  5865  cnvf1o  5972  brtpos2  5998  tposexg  6005  cnven  6505  cnvct  6506  fopwdom  6532  relcnvfi  6629
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