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Theorem cnvss 4426
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
cnvss (ABAB)

Proof of Theorem cnvss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 2910 . . . 4 (AB → (⟨y, x A → ⟨y, x B))
2 df-br 3731 . . . 4 (yAx ↔ ⟨y, x A)
3 df-br 3731 . . . 4 (yBx ↔ ⟨y, x B)
41, 2, 33imtr4g 194 . . 3 (AB → (yAxyBx))
54ssopab2dv 3981 . 2 (AB → {⟨x, y⟩ ∣ yAx} ⊆ {⟨x, y⟩ ∣ yBx})
6 df-cnv 4271 . 2 A = {⟨x, y⟩ ∣ yAx}
7 df-cnv 4271 . 2 B = {⟨x, y⟩ ∣ yBx}
85, 6, 73sstr4g 2957 1 (ABAB)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1369  wss 2888  cop 3345   class class class wbr 3730  {copab 3783  ccnv 4262
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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-in 2895  df-ss 2902  df-br 3731  df-opab 3785  df-cnv 4271
This theorem is referenced by:  cnveq  4427  rnss  4482  relcnvtr  4758  funss  4837  funcnvuni  4885  funres11  4888  funcnvres  4889  foimacnv  5060  tposss  5774
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