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 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))

Assertion

Ref	Expression
riotasv	⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 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

Step	Hyp	Ref	Expression
1		riotasv.1	. . 3 ⊢ 𝐴 ∈ V
2		riotasv.2	. . . . 5 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
3	2	a1i 11	. . . 4 ⊢ (𝐷 ∈ 𝐴 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
4		id 22	. . . 4 ⊢ (𝐷 ∈ 𝐴 → 𝐷 ∈ 𝐴)
5	3, 4	riotasvd 38957	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶))
6	1, 5	mpan2 691	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶))
7	6	3impib 1117	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)

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