ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elabrex Structured version   GIF version

Theorem elabrex 5318
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 2833 . . . . . . 7 (x = zB = z / xB)
32equcoms 1572 . . . . . 6 (z = xB = z / xB)
4 a1tru 1242 . . . . . 6 (z = x → ⊤ )
53, 42thd 164 . . . . 5 (z = x → (B = z / xB ↔ ⊤ ))
65rspcev 2629 . . . 4 ((x A ⊤ ) → z A B = z / xB)
71, 6mpan2 403 . . 3 (x Az A B = z / xB)
8 elabrex.1 . . . 4 B V
9 eqeq1 2024 . . . . 5 (y = B → (y = z / xBB = z / xB))
109rexbidv 2301 . . . 4 (y = B → (z A y = z / xBz A B = z / xB))
118, 10elab 2660 . . 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 1398 . . . 4 z y = B
14 nfcsb1v 2855 . . . . 5 xz / xB
1514nfeq2 2167 . . . 4 x y = z / xB
162eqeq2d 2029 . . . 4 (x = z → (y = By = z / xB))
1713, 15, 16cbvrex 2504 . . 3 (x A y = Bz A y = z / xB)
1817abbii 2131 . 2 {yx A y = B} = {yz A y = z / xB}
1912, 18syl6eleqr 2109 1 (x AB {yx A y = B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226  wtru 1227   wcel 1370  {cab 2004  wrex 2281  Vcvv 2531  csb 2825
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-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-csb 2826
This theorem is referenced by:  eusvobj2  5418
  Copyright terms: Public domain W3C validator