Theorem riotasv2d 34243

Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4874). Special case of riota2f 6632. (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 3212	. 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 2689	. . . . . . . 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 747	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐸 ∈ 𝐵 ∧ 𝜒)))
10		riotasv2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
11	10	eqeq2d 2632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝐷 = 𝐶 ↔ 𝐷 = 𝐹))
12	9, 11	imbi12d 334	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
13	12	adantlr 751	. . . 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 34242	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
17		riotasv2d.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
18		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
19	17, 18	nfan 1828	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐴 ∈ V)
20		nfcvd 2765	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦𝐸)
21		nfvd 1844	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦 𝐸 ∈ 𝐵)
22		riotasv2d.3	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
23	21, 22	nfand 1826	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ 𝜒))
24		nfra1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
25		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
26	24, 25	nfriota 6620	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
27	17, 14	nfceqdf 2760	. . . . . . . 8 ⊢ (𝜑 → (Ⅎ𝑦𝐷 ↔ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))))
28	26, 27	mpbiri 248	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐷)
29		riotasv2d.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐹)
30	28, 29	nfeqd 2772	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
31	23, 30	nfimd 1823	. . . . 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 3256	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
34	3, 5, 33	mp2and 715	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐷 = 𝐹)
35	1, 34	sylan2 491	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  ∀wral 2912  Vcvv 3200  ℩crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-undef 7399

This theorem is referenced by:  riotasv2s  34244  cdleme42b  35766