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 37450
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5363). Special case of riota2f 7343. (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 37447 . . 3 ((𝐷𝐴𝐴 ∈ V) → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
61, 5mpan2 690 . 2 (𝐷𝐴 → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
763impib 1117 1 ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  crio 7317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-riotaBAD 37444
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-undef 8209
This theorem is referenced by:  cdleme26e  38851  cdleme26eALTN  38853  cdleme26fALTN  38854  cdleme26f  38855  cdleme26f2ALTN  38856  cdleme26f2  38857
  Copyright terms: Public domain W3C validator