Theorem riotasv2d 36971

Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5326). Special case of riota2f 7257. (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 3450	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv2d.8	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐸 ∈ 𝐵)
4		riotasv2d.9	. . . 4 ⊢ (𝜑 → 𝜒)
5	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜒)
6		eleq1 2826	. . . . . . . 8 ⊢ (𝑦 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
7	6	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
8		riotasv2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐸 ∈ 𝐵 ∧ 𝜒)))
10		riotasv2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
11	10	eqeq2d 2749	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝐷 = 𝐶 ↔ 𝐷 = 𝐹))
12	9, 11	imbi12d 345	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
13	12	adantlr 712	. . . 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 36970	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
17		riotasv2d.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
18		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
19	17, 18	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐴 ∈ V)
20		nfcvd 2908	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦𝐸)
21		nfvd 1918	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦 𝐸 ∈ 𝐵)
22		riotasv2d.3	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
23	21, 22	nfand 1900	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ 𝜒))
24		nfra1 3144	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
25		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
26	24, 25	nfriota 7245	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
27	17, 14	nfceqdf 2902	. . . . . . . 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 2917	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
31	23, 30	nfimd 1897	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
32	31	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
33	3, 13, 16, 19, 20, 32	vtocldf 3493	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
34	3, 5, 33	mp2and 696	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐷 = 𝐹)
35	1, 34	sylan2 593	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  Vcvv 3432  ℩crio 7231

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-undef 8089

This theorem is referenced by:  riotasv2s  36972  cdleme42b  38492