Theorem riotasv2d 40108

Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5364). Special case of riota2f 7402. (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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv2d.8	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐸 ∈ 𝐵)
4		riotasv2d.9	. . . 4 ⊢ (𝜑 → 𝜒)
5	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜒)
6		eleq1 2848	. . . . . . . 8 ⊢ (𝑦 = 𝐸 → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
7	6	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝑦 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
8		riotasv2d.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))
9	7, 8	anbi12d 644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐸 ∈ 𝐵 ∧ 𝜒)))
10		riotasv2d.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)
11	10	eqeq2d 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝐷 = 𝐶 ↔ 𝐷 = 𝐹))
12	9, 11	imbi12d 347	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶) ↔ ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹)))
13	12	adantlr 728	. . . 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 40107	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
17		riotasv2d.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
18		nfv 1946	. . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V
19	17, 18	nfan 1931	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐴 ∈ V)
20		nfcvd 2923	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦𝐸)
21		nfvd 1947	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦 𝐸 ∈ 𝐵)
22		riotasv2d.3	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
23	21, 22	nfand 1929	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ 𝜒))
24		nfra1 3286	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
25		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
26	24, 25	nfriota 7390	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
27	17, 14	nfceqdf 2918	. . . . . . . 8 ⊢ (𝜑 → (Ⅎ𝑦𝐷 ↔ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))))
28	26, 27	mpbiri 261	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐷)
29		riotasv2d.2	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝐹)
30	28, 29	nfeqd 2932	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
31	23, 30	nfimd 1926	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
32	31	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → Ⅎ𝑦((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
33	3, 13, 16, 19, 20, 32	vtocldf 3521	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝐸 ∈ 𝐵 ∧ 𝜒) → 𝐷 = 𝐹))
34	3, 5, 33	mp2and 712	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝐷 = 𝐹)
35	1, 34	sylan2 605	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  Vcvv 3450  ℩crio 7377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-riotaBAD 40104

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-riota 7378  df-undef 8294

This theorem is used by:  riotasv2s  40109  cdleme42b  41629