Theorem reusv2 5269

Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦 ∈ 𝐵 such that 𝜑. The first antecedent ensures that the constant value belongs to the existential uniqueness domain 𝐴, and the second ensures that 𝐶(𝑦) is evaluated for at least one 𝑦. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2	⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrab1 3337	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
2		nfcv 2955	. . . 4 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝐶 ∈ 𝐴
4		nfcsb1v 3852	. . . . 5 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
5	4	nfel1 2971	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴
6		csbeq1a 3842	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
7	6	eleq1d 2874	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴))
8	1, 2, 3, 5, 7	cbvralfw 3382	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴)
9		rabid 3331	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
10	9	imbi1i 353	. . . . 5 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴))
11		impexp 454	. . . . 5 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
12	10, 11	bitri 278	. . . 4 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
13	12	ralbii2 3131	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
14	8, 13	bitr3i 280	. 2 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
15		rabn0 4293	. 2 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
16		reusv2lem5 5268	. . 3 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
17		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝐶
18	4	nfeq2 2972	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19	6	eqeq2d 2809	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
20	1, 2, 17, 18, 19	cbvrexfw 3384	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
21	9	anbi1i 626	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶))
22		anass 472	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
23	21, 22	bitri 278	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
24	23	rexbii2 3208	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
25	20, 24	bitr3i 280	. . . 4 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
26	25	reubii 3344	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
27	1, 2, 17, 18, 19	cbvralfw 3382	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
28	9	imbi1i 353	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶))
29		impexp 454	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
30	28, 29	bitri 278	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
31	30	ralbii2 3131	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
32	27, 31	bitr3i 280	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
33	32	reubii 3344	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
34	16, 26, 33	3bitr3g 316	. 2 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
35	14, 15, 34	syl2anbr 601	1 ⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  {crab 3110  ⦋csb 3828  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-nul 4244

This theorem is referenced by:  cdleme25dN  37652