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Theorem cnvss 4431
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 2912 . . . 4 (AB → (⟨y, x A → ⟨y, x B))
2 df-br 3735 . . . 4 (yAx ↔ ⟨y, x A)
3 df-br 3735 . . . 4 (yBx ↔ ⟨y, x B)
41, 2, 33imtr4g 194 . . 3 (AB → (yAxyBx))
54ssopab2dv 3985 . 2 (AB → {⟨x, y⟩ ∣ yAx} ⊆ {⟨x, y⟩ ∣ yBx})
6 df-cnv 4276 . 2 A = {⟨x, y⟩ ∣ yAx}
7 df-cnv 4276 . 2 B = {⟨x, y⟩ ∣ yBx}
85, 6, 73sstr4g 2959 1 (ABAB)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1370  wss 2890  cop 3349   class class class wbr 3734  {copab 3787  ccnv 4267
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-in 2897  df-ss 2904  df-br 3735  df-opab 3789  df-cnv 4276
This theorem is referenced by:  cnveq  4432  rnss  4487  relcnvtr  4763  funss  4842  funcnvuni  4890  funres11  4893  funcnvres  4894  foimacnv  5065  tposss  5779
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