Theorem riotacl2 7419

Description: Membership law for "the unique element in 𝐴 such that 𝜑". (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
riotacl2	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

Proof of Theorem riotacl2

Step	Hyp	Ref	Expression
1		df-reu 3370	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iotacl 6546	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
3	1, 2	sylbi 220	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4		df-riota 7403	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rab 3417	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	3, 4, 5	3eltr4g 2881	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∃!weu 2596  {cab 2741  ∃!wreu 3367  {crab 3416  ℩cio 6513  ℩crio 7402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6515  df-riota 7403

This theorem is used by:  riotacl  7420  riotasbc  7421  riotaxfrd  7437  supub  9481  suplub  9482  ordtypelem3  9544  catlid  17889  catrid  17890  grplinv  19232  pj1id  19946  evlsval2  22429  ig1pval3  26529  coelem  26578  quotlem  26654  mircgr  29162  mirbtwn  29163  grpoidinv2  31150  grpoinv  31160  cnlnadjlem5  32706  cvmsiota  36222  cvmliftiota  36246  weiunlem  37442  weiunfrlem  37443  linvh  43349  mpaalem  44357  disjinfi  46422