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Theorem elabrex 5313
 Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
Hypothesis
Ref Expression
elabrex.1 B V
Assertion
Ref Expression
elabrex (x AB {yx A y = B})
Distinct variable groups:   y,B   x,y,A
Allowed substitution hint:   B(x)

Proof of Theorem elabrex
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 tru 1230 . . . 4
2 csbeq1a 2831 . . . . . . 7 (x = zB = z / xB)
32equcoms 1570 . . . . . 6 (z = xB = z / xB)
4 a1tru 1242 . . . . . 6 (z = x → ⊤ )
53, 42thd 164 . . . . 5 (z = x → (B = z / xB ↔ ⊤ ))
65rspcev 2627 . . . 4 ((x A ⊤ ) → z A B = z / xB)
71, 6mpan2 401 . . 3 (x Az A B = z / xB)
8 elabrex.1 . . . 4 B V
9 eqeq1 2022 . . . . 5 (y = B → (y = z / xBB = z / xB))
109rexbidv 2299 . . . 4 (y = B → (z A y = z / xBz A B = z / xB))
118, 10elab 2658 . . 3 (B {yz A y = z / xB} ↔ z A B = z / xB)
127, 11sylibr 137 . 2 (x AB {yz A y = z / xB})
13 nfv 1397 . . . 4 z y = B
14 nfcsb1v 2853 . . . . 5 xz / xB
1514nfeq2 2165 . . . 4 x y = z / xB
162eqeq2d 2027 . . . 4 (x = z → (y = By = z / xB))
1713, 15, 16cbvrex 2502 . . 3 (x A y = Bz A y = z / xB)
1817abbii 2129 . 2 {yx A y = B} = {yz A y = z / xB}
1912, 18syl6eleqr 2107 1 (x AB {yx A y = B})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1226   ⊤ wtru 1227   ∈ wcel 1369  {cab 2002  ∃wrex 2279  Vcvv 2529  ⦋csb 2823 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-tru 1229  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-rex 2284  df-v 2531  df-sbc 2736  df-csb 2824 This theorem is referenced by:  eusvobj2  5413
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