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Theorem elabrex 5611
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
Hypothesis
Ref Expression
elabrex.1 𝐵 ∈ V
Assertion
Ref Expression
elabrex (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 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 1316 . . . 4
2 csbeq1a 2977 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 1665 . . . . . 6 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 a1tru 1328 . . . . . 6 (𝑧 = 𝑥 → ⊤)
53, 42thd 174 . . . . 5 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 2758 . . . 4 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 419 . . 3 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
8 elabrex.1 . . . 4 𝐵 ∈ V
9 eqeq1 2119 . . . . 5 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 2410 . . . 4 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
118, 10elab 2796 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
127, 11sylibr 133 . 2 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
13 nfv 1489 . . . 4 𝑧 𝑦 = 𝐵
14 nfcsb1v 2999 . . . . 5 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2265 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
162eqeq2d 2124 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1713, 15, 16cbvrex 2623 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1817abbii 2228 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
1912, 18syl6eleqr 2206 1 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1312  wtru 1313  wcel 1461  {cab 2099  wrex 2389  Vcvv 2655  csb 2969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-sbc 2877  df-csb 2970
This theorem is referenced by:  eusvobj2  5712
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