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Theorem cnvexg 5165
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 5005 . . 3 Rel 𝐴
2 relssdmrn 5148 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4636 . . . 4 ran 𝐴 = dom 𝐴
5 rnexg 4891 . . . 4 (𝐴𝑉 → ran 𝐴 ∈ V)
64, 5eqeltrrid 2265 . . 3 (𝐴𝑉 → dom 𝐴 ∈ V)
7 dfdm4 4818 . . . 4 dom 𝐴 = ran 𝐴
8 dmexg 4890 . . . 4 (𝐴𝑉 → dom 𝐴 ∈ V)
97, 8eqeltrrid 2265 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
10 xpexg 4739 . . 3 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 × ran 𝐴) ∈ V)
116, 9, 10syl2anc 411 . 2 (𝐴𝑉 → (dom 𝐴 × ran 𝐴) ∈ V)
12 ssexg 4141 . 2 ((𝐴 ⊆ (dom 𝐴 × ran 𝐴) ∧ (dom 𝐴 × ran 𝐴) ∈ V) → 𝐴 ∈ V)
133, 11, 12sylancr 414 1 (𝐴𝑉𝐴 ∈ 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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