ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvexg GIF version

Theorem cnvexg 4882
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 4730 . . 3 Rel 𝐴
2 relssdmrn 4868 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 7 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4383 . . . 4 ran 𝐴 = dom 𝐴
5 rnexg 4624 . . . 4 (𝐴𝑉 → ran 𝐴 ∈ V)
64, 5syl5eqelr 2141 . . 3 (𝐴𝑉 → dom 𝐴 ∈ V)
7 dfdm4 4554 . . . 4 dom 𝐴 = ran 𝐴
8 dmexg 4623 . . . 4 (𝐴𝑉 → dom 𝐴 ∈ V)
97, 8syl5eqelr 2141 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
10 xpexg 4479 . . 3 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 × ran 𝐴) ∈ V)
116, 9, 10syl2anc 397 . 2 (𝐴𝑉 → (dom 𝐴 × ran 𝐴) ∈ V)
12 ssexg 3923 . 2 ((𝐴 ⊆ (dom 𝐴 × ran 𝐴) ∧ (dom 𝐴 × ran 𝐴) ∈ V) → 𝐴 ∈ V)
133, 11, 12sylancr 399 1 (𝐴𝑉𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  Vcvv 2574  wss 2944   × cxp 4370  ccnv 4371  dom cdm 4372  ran crn 4373  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  cnvex  4883  relcnvexb  4884  cofunex2g  5766  cnvf1o  5873  brtpos2  5896  tposexg  5903  cnven  6318  fopwdom  6340
  Copyright terms: Public domain W3C validator