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Theorem elabrex 7243
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 1543 . . . 4
2 csbeq1a 3906 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 2021 . . . . . 6 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 trud 1549 . . . . . 6 (𝑧 = 𝑥 → ⊤)
53, 42thd 264 . . . . 5 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3611 . . . 4 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 687 . . 3 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
8 elabrex.1 . . . 4 𝐵 ∈ V
9 eqeq1 2734 . . . . 5 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3176 . . . 4 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
118, 10elab 3667 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
127, 11sylibr 233 . 2 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
13 nfv 1915 . . . 4 𝑧 𝑦 = 𝐵
14 nfcsb1v 3917 . . . . 5 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2918 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
162eqeq2d 2741 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1713, 15, 16cbvrexw 3302 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1817abbii 2800 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
1912, 18eleqtrrdi 2842 1 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wtru 1540  wcel 2104  {cab 2707  wrex 3068  Vcvv 3472  csb 3892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-sbc 3777  df-csb 3893
This theorem is referenced by:  eusvobj2  7403  lss1d  20718  prdsxmetlem  24094  prdsbl  24220  itg2monolem1  25500  heibor1  36981  dihglblem5  40472
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