Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  riotaclbgBAD Structured version   Visualization version   GIF version

Theorem riotaclbgBAD 34933
Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotaclbgBAD (𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem riotaclbgBAD
StepHypRef Expression
1 riotacl 6821 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
2 undefnel2 7610 . . . 4 (𝐴𝑉 → ¬ (Undef‘𝐴) ∈ 𝐴)
3 iffalse 4254 . . . . . . 7 (¬ ∃!𝑥𝐴 𝜑 → if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴})) = (Undef‘{𝑥𝑥𝐴}))
4 ax-riotaBAD 34932 . . . . . . 7 (𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))
5 abid1 2887 . . . . . . . 8 𝐴 = {𝑥𝑥𝐴}
65fveq2i 6382 . . . . . . 7 (Undef‘𝐴) = (Undef‘{𝑥𝑥𝐴})
73, 4, 63eqtr4g 2824 . . . . . 6 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = (Undef‘𝐴))
87eleq1d 2829 . . . . 5 (¬ ∃!𝑥𝐴 𝜑 → ((𝑥𝐴 𝜑) ∈ 𝐴 ↔ (Undef‘𝐴) ∈ 𝐴))
98notbid 309 . . . 4 (¬ ∃!𝑥𝐴 𝜑 → (¬ (𝑥𝐴 𝜑) ∈ 𝐴 ↔ ¬ (Undef‘𝐴) ∈ 𝐴))
102, 9syl5ibrcom 238 . . 3 (𝐴𝑉 → (¬ ∃!𝑥𝐴 𝜑 → ¬ (𝑥𝐴 𝜑) ∈ 𝐴))
1110con4d 115 . 2 (𝐴𝑉 → ((𝑥𝐴 𝜑) ∈ 𝐴 → ∃!𝑥𝐴 𝜑))
121, 11impbid2 217 1 (𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wcel 2155  {cab 2751  ∃!wreu 3057  ifcif 4245  cio 6031  cfv 6070  crio 6806  Undefcund 7605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-riotaBAD 34932
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-riota 6807  df-undef 7606
This theorem is referenced by:  riotaclbBAD  34934  riotasvd  34935
  Copyright terms: Public domain W3C validator