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Theorem elabrexg 42048
 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 1542 . . . . 5
2 csbeq1a 3819 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 2027 . . . . . . 7 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 trud 1548 . . . . . . 7 (𝑧 = 𝑥 → ⊤)
53, 42thd 268 . . . . . 6 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3541 . . . . 5 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 690 . . . 4 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
87adantr 484 . . 3 ((𝑥𝐴𝐵𝑉) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
9 eqeq1 2762 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3221 . . . . 5 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1110elabg 3587 . . . 4 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1211adantl 485 . . 3 ((𝑥𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
138, 12mpbird 260 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
14 nfv 1915 . . . 4 𝑧 𝑦 = 𝐵
15 nfcsb1v 3829 . . . . 5 𝑥𝑧 / 𝑥𝐵
1615nfeq2 2936 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
172eqeq2d 2769 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1814, 16, 17cbvrexw 3353 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1918abbii 2823 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
2013, 19eleqtrrdi 2863 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {cab 2735  ∃wrex 3071  ⦋csb 3805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-sbc 3697  df-csb 3806 This theorem is referenced by:  upbdrech  42305  ssfiunibd  42309
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