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

Theorem rabsnif 4478
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
Hypothesis
Ref Expression
rabsnif.f (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rabsnif {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnif
StepHypRef Expression
1 rabsnifsb 4477 . . 3 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2 rabsnif.f . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3695 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜓))
43ifbid 4330 . . 3 (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
51, 4syl5eq 2873 . 2 (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6 rab0 4187 . . . 4 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7 ifid 4347 . . . 4 if(𝜓, ∅, ∅) = ∅
86, 7eqtr4i 2852 . . 3 {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9 snprc 4473 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 208 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
1110rabeqdv 3407 . . 3 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
1210ifeq1d 4326 . . 3 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
138, 11, 123eqtr4a 2887 . 2 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
145, 13pm2.61i 177 1 {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  [wsbc 3662  c0 4146  ifcif 4308  {csn 4399
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-3an 1113  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-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-nul 4147  df-if 4309  df-sn 4400
This theorem is referenced by:  suppsnop  7578  m1detdiag  20778  1loopgrvd2  26808  1hevtxdg1  26811  1egrvtxdg1  26814
  Copyright terms: Public domain W3C validator