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Theorem elabrexg 38724
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 1484 . . . . 5
2 csbeq1a 3527 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 1944 . . . . . . 7 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 a1tru 1497 . . . . . . 7 (𝑧 = 𝑥 → ⊤)
53, 42thd 255 . . . . . 6 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3298 . . . . 5 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 706 . . . 4 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
87adantr 481 . . 3 ((𝑥𝐴𝐵𝑉) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
9 eqeq1 2625 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3046 . . . . 5 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1110elabg 3338 . . . 4 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
1211adantl 482 . . 3 ((𝑥𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
138, 12mpbird 247 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
14 nfv 1840 . . . 4 𝑧 𝑦 = 𝐵
15 nfcsb1v 3534 . . . . 5 𝑥𝑧 / 𝑥𝐵
1615nfeq2 2776 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
172eqeq2d 2631 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1814, 16, 17cbvrex 3159 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1918abbii 2736 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
2013, 19syl6eleqr 2709 1 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  {cab 2607  wrex 2908  csb 3518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-sbc 3422  df-csb 3519
This theorem is referenced by:  upbdrech  39014  ssfiunibd  39018
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