Theorem riotasv2d 38556

Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5403). Special case of riota2f 7400. (Contributed by NM, 2-Mar-2013.)

Hypotheses

Ref	Expression
riotasv2d.1	⊢ Ⅎ𝑦𝜑
riotasv2d.2	⊢ (𝜑 → Ⅎ𝑦𝐹)
riotasv2d.3	⊢ (𝜑 → Ⅎ𝑦𝜒)
riotasv2d.4	⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
riotasv2d.5	⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))
riotasv2d.6	⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
riotasv2d.7	⊢ (𝜑 → 𝐷 ∈ 𝐴)
riotasv2d.8	⊢ (𝜑 → 𝐸 ∈ 𝐵)
riotasv2d.9	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
riotasv2d	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑦,𝐸   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2d

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv2d.8	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3	2	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐸 ∈ 𝐵)
4		riotasv2d.9	. . . 4 ⊢ (𝜑 → 𝜒)
5	4	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜒)
6		eleq1 2813	. . . . . . . 8 ⊢ (𝑦 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
7	6	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
8		riotasv2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐸 ∈ 𝐵 ∧ 𝜒)))
10		riotasv2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
11	10	eqeq2d 2736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝐷 = 𝐶 ↔ 𝐷 = 𝐹))
12	9, 11	imbi12d 343	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
13	12	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
14		riotasv2d.4	. . . . 5 ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
15		riotasv2d.7	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
16	14, 15	riotasvd 38555	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
17		riotasv2d.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
18		nfv 1909	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
19	17, 18	nfan 1894	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐴 ∈ V)
20		nfcvd 2892	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦𝐸)
21		nfvd 1910	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦 𝐸 ∈ 𝐵)
22		riotasv2d.3	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
23	21, 22	nfand 1892	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ 𝜒))
24		nfra1 3271	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
25		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
26	24, 25	nfriota 7388	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
27	17, 14	nfceqdf 2886	. . . . . . . 8 ⊢ (𝜑 → (Ⅎ𝑦𝐷 ↔ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))))
28	26, 27	mpbiri 257	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐷)
29		riotasv2d.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐹)
30	28, 29	nfeqd 2902	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
31	23, 30	nfimd 1889	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
32	31	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
33	3, 13, 16, 19, 20, 32	vtocldf 3537	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
34	3, 5, 33	mp2and 697	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐷 = 𝐹)
35	1, 34	sylan2 591	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3050  Vcvv 3461  ℩crio 7374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-riotaBAD 38552

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-riota 7375  df-undef 8279

This theorem is referenced by:  riotasv2s  38557  cdleme42b  40078