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Theorem elabrex 5340
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 1246 . . . 4
2 csbeq1a 2854 . . . . . . 7 (x = zB = z / xB)
32equcoms 1591 . . . . . 6 (z = xB = z / xB)
4 a1tru 1258 . . . . . 6 (z = x → ⊤ )
53, 42thd 164 . . . . 5 (z = x → (B = z / xB ↔ ⊤ ))
65rspcev 2650 . . . 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 2043 . . . . 5 (y = B → (y = z / xBB = z / xB))
109rexbidv 2321 . . . 4 (y = B → (z A y = z / xBz A B = z / xB))
118, 10elab 2681 . . 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 1418 . . . 4 z y = B
14 nfcsb1v 2876 . . . . 5 xz / xB
1514nfeq2 2186 . . . 4 x y = z / xB
162eqeq2d 2048 . . . 4 (x = z → (y = By = z / xB))
1713, 15, 16cbvrex 2524 . . 3 (x A y = Bz A y = z / xB)
1817abbii 2150 . 2 {yx A y = B} = {yz A y = z / xB}
1912, 18syl6eleqr 2128 1 (x AB {yx A y = B})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wtru 1243   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551  csb 2846
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  eusvobj2  5441
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