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

Theorem riotasv 38960
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5403). Special case of riota2f 7412. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
riotasv.1 𝐴 ∈ V
riotasv.2 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
Assertion
Ref Expression
riotasv ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem riotasv
StepHypRef Expression
1 riotasv.1 . . 3 𝐴 ∈ V
2 riotasv.2 . . . . 5 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
32a1i 11 . . . 4 (𝐷𝐴𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
4 id 22 . . . 4 (𝐷𝐴𝐷𝐴)
53, 4riotasvd 38957 . . 3 ((𝐷𝐴𝐴 ∈ V) → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
61, 5mpan2 691 . 2 (𝐷𝐴 → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
763impib 1117 1 ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-undef 8298
This theorem is referenced by:  cdleme26e  40361  cdleme26eALTN  40363  cdleme26fALTN  40364  cdleme26f  40365  cdleme26f2ALTN  40366  cdleme26f2  40367
  Copyright terms: Public domain W3C validator