Theorem reusv2 5400

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 3451	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
2		nfcv 2903	. . . 4 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
3		nfv 1917	. . . 4 ⊢ Ⅎ𝑧 𝐶 ∈ 𝐴
4		nfcsb1v 3917	. . . . 5 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
5	4	nfel1 2919	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴
6		csbeq1a 3906	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
7	6	eleq1d 2818	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴))
8	1, 2, 3, 5, 7	cbvralfw 3301	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴)
9		rabid 3452	. . . . . 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 451	. . . . 5 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
12	10, 11	bitri 274	. . . 4 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
13	12	ralbii2 3089	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
14	8, 13	bitr3i 276	. 2 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
15		rabn0 4384	. 2 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
16		reusv2lem5 5399	. . 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 1917	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝐶
18	4	nfeq2 2920	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19	6	eqeq2d 2743	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
20	1, 2, 17, 18, 19	cbvrexfw 3302	. . . . 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 469	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
23	21, 22	bitri 274	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
24	23	rexbii2 3090	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
25	20, 24	bitr3i 276	. . . 4 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
26	25	reubii 3385	. . 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 3301	. . . . 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 451	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
30	28, 29	bitri 274	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
31	30	ralbii2 3089	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
32	27, 31	bitr3i 276	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
33	32	reubii 3385	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
34	16, 26, 33	3bitr3g 312	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∃!wreu 3374  {crab 3432  ⦋csb 3892  ∅c0 4321

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305  ax-pow 5362

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-nul 4322

This theorem is referenced by:  cdleme25dN  39215