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Theorem cnvexg 4798
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 (A 𝑉A V)

Proof of Theorem cnvexg
StepHypRef Expression
1 relcnv 4646 . . 3 Rel A
2 relssdmrn 4784 . . 3 (Rel AA ⊆ (dom A × ran A))
31, 2ax-mp 7 . 2 A ⊆ (dom A × ran A)
4 df-rn 4299 . . . 4 ran A = dom A
5 rnexg 4540 . . . 4 (A 𝑉 → ran A V)
64, 5syl5eqelr 2122 . . 3 (A 𝑉 → dom A V)
7 dfdm4 4470 . . . 4 dom A = ran A
8 dmexg 4539 . . . 4 (A 𝑉 → dom A V)
97, 8syl5eqelr 2122 . . 3 (A 𝑉 → ran A V)
10 xpexg 4395 . . 3 ((dom A V ran A V) → (dom A × ran A) V)
116, 9, 10syl2anc 391 . 2 (A 𝑉 → (dom A × ran A) V)
12 ssexg 3887 . 2 ((A ⊆ (dom A × ran A) (dom A × ran A) V) → A V)
133, 11, 12sylancr 393 1 (A 𝑉A V)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  Vcvv 2551  wss 2911   × cxp 4286  ccnv 4287  dom cdm 4288  ran crn 4289  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  cnvex  4799  relcnvexb  4800  cofunex2g  5681  cnvf1o  5788  brtpos2  5807  tposexg  5814  cnven  6224  fopwdom  6246
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