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Theorem cnvexg 5044
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 4885 . . 3 Rel 𝐴
2 relssdmrn 5027 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4518 . . . 4 ran 𝐴 = dom 𝐴
5 rnexg 4772 . . . 4 (𝐴𝑉 → ran 𝐴 ∈ V)
64, 5eqeltrrid 2203 . . 3 (𝐴𝑉 → dom 𝐴 ∈ V)
7 dfdm4 4699 . . . 4 dom 𝐴 = ran 𝐴
8 dmexg 4771 . . . 4 (𝐴𝑉 → dom 𝐴 ∈ V)
97, 8eqeltrrid 2203 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
10 xpexg 4621 . . 3 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 × ran 𝐴) ∈ V)
116, 9, 10syl2anc 406 . 2 (𝐴𝑉 → (dom 𝐴 × ran 𝐴) ∈ V)
12 ssexg 4035 . 2 ((𝐴 ⊆ (dom 𝐴 × ran 𝐴) ∧ (dom 𝐴 × ran 𝐴) ∈ V) → 𝐴 ∈ V)
133, 11, 12sylancr 408 1 (𝐴𝑉𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  Vcvv 2658  wss 3039   × cxp 4505  ccnv 4506  dom cdm 4507  ran crn 4508  Rel wrel 4512
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  cnvex  5045  relcnvexb  5046  cofunex2g  5976  cnvf1o  6088  brtpos2  6114  tposexg  6121  cnven  6668  cnvct  6669  fopwdom  6696  relcnvfi  6795  ennnfonelemim  11843
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