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Theorem elabrexg 43728
Description: Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elabrexg ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabrexg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tru 1546 . . . . 5
2 csbeq1a 3908 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 2024 . . . . . . 7 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 trud 1552 . . . . . . 7 (𝑧 = 𝑥 → ⊤)
53, 42thd 265 . . . . . 6 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3613 . . . . 5 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 690 . . . 4 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
87adantr 482 . . 3 ((𝑥𝐴𝐵𝑉) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
9 eqeq1 2737 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3179 . . . . 5 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1110elabg 3667 . . . 4 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1211adantl 483 . . 3 ((𝑥𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
138, 12mpbird 257 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
14 nfv 1918 . . . 4 𝑧 𝑦 = 𝐵
15 nfcsb1v 3919 . . . . 5 𝑥𝑧 / 𝑥𝐵
1615nfeq2 2921 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
172eqeq2d 2744 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1814, 16, 17cbvrexw 3305 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1918abbii 2803 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
2013, 19eleqtrrdi 2845 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wtru 1543  wcel 2107  {cab 2710  wrex 3071  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-sbc 3779  df-csb 3895
This theorem is referenced by:  upbdrech  44015  ssfiunibd  44019
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