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Theorem elabrex 6993
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 1532 . . . 4
2 csbeq1a 3894 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 2018 . . . . . 6 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 trud 1538 . . . . . 6 (𝑧 = 𝑥 → ⊤)
53, 42thd 266 . . . . 5 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3620 . . . 4 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 687 . . 3 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
8 elabrex.1 . . . 4 𝐵 ∈ V
9 eqeq1 2822 . . . . 5 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3294 . . . 4 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
118, 10elab 3664 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
127, 11sylibr 235 . 2 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
13 nfv 1906 . . . 4 𝑧 𝑦 = 𝐵
14 nfcsb1v 3904 . . . . 5 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2992 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
162eqeq2d 2829 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1713, 15, 16cbvrexw 3440 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1817abbii 2883 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
1912, 18eleqtrrdi 2921 1 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wtru 1529  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  csb 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-csb 3881
This theorem is referenced by:  eusvobj2  7138  lss1d  19664  prdsxmetlem  22905  prdsbl  23028  itg2monolem1  24278  heibor1  34969  dihglblem5  38314
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