Theorem riotasv2d 39545

Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5359). Special case of riota2f 7373. (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 3474	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv2d.8	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐸 ∈ 𝐵)
4		riotasv2d.9	. . . 4 ⊢ (𝜑 → 𝜒)
5	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜒)
6		eleq1 2849	. . . . . . . 8 ⊢ (𝑦 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
7	6	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
8		riotasv2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐸 ∈ 𝐵 ∧ 𝜒)))
10		riotasv2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
11	10	eqeq2d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝐷 = 𝐶 ↔ 𝐷 = 𝐹))
12	9, 11	imbi12d 346	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
13	12	adantlr 725	. . . 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 39544	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
17		riotasv2d.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
18		nfv 1933	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
19	17, 18	nfan 1918	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐴 ∈ V)
20		nfcvd 2924	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦𝐸)
21		nfvd 1934	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦 𝐸 ∈ 𝐵)
22		riotasv2d.3	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
23	21, 22	nfand 1916	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ 𝜒))
24		nfra1 3285	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
25		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
26	24, 25	nfriota 7361	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
27	17, 14	nfceqdf 2919	. . . . . . . 8 ⊢ (𝜑 → (Ⅎ𝑦𝐷 ↔ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))))
28	26, 27	mpbiri 260	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐷)
29		riotasv2d.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐹)
30	28, 29	nfeqd 2933	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
31	23, 30	nfimd 1913	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
32	31	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
33	3, 13, 16, 19, 20, 32	vtocldf 3526	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
34	3, 5, 33	mp2and 709	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐷 = 𝐹)
35	1, 34	sylan2 602	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141  Ⅎwnfc 2908  ∀wral 3075  Vcvv 3453  ℩crio 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-riotaBAD 39541

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-riota 7349  df-undef 8248

This theorem is referenced by:  riotasv2s  39546  cdleme42b  41066