Theorem reusv2 5409

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 3454	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
2		nfcv 2903	. . . 4 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
3		nfv 1912	. . . 4 ⊢ Ⅎ𝑧 𝐶 ∈ 𝐴
4		nfcsb1v 3933	. . . . 5 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
5	4	nfel1 2920	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴
6		csbeq1a 3922	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
7	6	eleq1d 2824	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴))
8	1, 2, 3, 5, 7	cbvralfw 3302	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴)
9		rabid 3455	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
10	9	imbi1i 349	. . . . 5 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴))
11		impexp 450	. . . . 5 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
12	10, 11	bitri 275	. . . 4 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
13	12	ralbii2 3087	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
14	8, 13	bitr3i 277	. 2 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
15		rabn0 4395	. 2 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
16		reusv2lem5 5408	. . 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 1912	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝐶
18	4	nfeq2 2921	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19	6	eqeq2d 2746	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
20	1, 2, 17, 18, 19	cbvrexfw 3303	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
21	9	anbi1i 624	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶))
22		anass 468	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
23	21, 22	bitri 275	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
24	23	rexbii2 3088	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
25	20, 24	bitr3i 277	. . . 4 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
26	25	reubii 3387	. . 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 3302	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
28	9	imbi1i 349	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶))
29		impexp 450	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
30	28, 29	bitri 275	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
31	30	ralbii2 3087	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
32	27, 31	bitr3i 277	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
33	32	reubii 3387	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
34	16, 26, 33	3bitr3g 313	. 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 599	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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376  {crab 3433  ⦋csb 3908  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pow 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340

This theorem is referenced by:  cdleme25dN  40339