MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elabrex Structured version   Visualization version   GIF version

Theorem elabrex 6761
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 1661 . . . 4
2 csbeq1a 3766 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
32equcoms 2124 . . . . . 6 (𝑧 = 𝑥𝐵 = 𝑧 / 𝑥𝐵)
4 trud 1667 . . . . . 6 (𝑧 = 𝑥 → ⊤)
53, 42thd 257 . . . . 5 (𝑧 = 𝑥 → (𝐵 = 𝑧 / 𝑥𝐵 ↔ ⊤))
65rspcev 3526 . . . 4 ((𝑥𝐴 ∧ ⊤) → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
71, 6mpan2 682 . . 3 (𝑥𝐴 → ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
8 elabrex.1 . . . 4 𝐵 ∈ V
9 eqeq1 2829 . . . . 5 (𝑦 = 𝐵 → (𝑦 = 𝑧 / 𝑥𝐵𝐵 = 𝑧 / 𝑥𝐵))
109rexbidv 3262 . . . 4 (𝑦 = 𝐵 → (∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵 ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵))
118, 10elab 3571 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵} ↔ ∃𝑧𝐴 𝐵 = 𝑧 / 𝑥𝐵)
127, 11sylibr 226 . 2 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵})
13 nfv 2013 . . . 4 𝑧 𝑦 = 𝐵
14 nfcsb1v 3773 . . . . 5 𝑥𝑧 / 𝑥𝐵
1514nfeq2 2985 . . . 4 𝑥 𝑦 = 𝑧 / 𝑥𝐵
162eqeq2d 2835 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝐵𝑦 = 𝑧 / 𝑥𝐵))
1713, 15, 16cbvrex 3380 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵)
1817abbii 2944 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧𝐴 𝑦 = 𝑧 / 𝑥𝐵}
1912, 18syl6eleqr 2917 1 (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wtru 1657  wcel 2164  {cab 2811  wrex 3118  Vcvv 3414  csb 3757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-sbc 3663  df-csb 3758
This theorem is referenced by:  eusvobj2  6903  lss1d  19329  prdsxmetlem  22550  prdsbl  22673  itg2monolem1  23923  heibor1  34150  dihglblem5  37372
  Copyright terms: Public domain W3C validator