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Theorem cnvss 4451
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 2933 . . . 4 (AB → (⟨y, x A → ⟨y, x B))
2 df-br 3756 . . . 4 (yAx ↔ ⟨y, x A)
3 df-br 3756 . . . 4 (yBx ↔ ⟨y, x B)
41, 2, 33imtr4g 194 . . 3 (AB → (yAxyBx))
54ssopab2dv 4006 . 2 (AB → {⟨x, y⟩ ∣ yAx} ⊆ {⟨x, y⟩ ∣ yBx})
6 df-cnv 4296 . 2 A = {⟨x, y⟩ ∣ yAx}
7 df-cnv 4296 . 2 B = {⟨x, y⟩ ∣ yBx}
85, 6, 73sstr4g 2980 1 (ABAB)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  {copab 3808  ccnv 4287
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by:  cnveq  4452  rnss  4507  relcnvtr  4783  funss  4863  funcnvuni  4911  funres11  4914  funcnvres  4915  foimacnv  5087  tposss  5802
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