Theorem reusv2 5359

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 3433	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
2		nfcv 2923	. . . 4 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
3		nfv 1933	. . . 4 ⊢ Ⅎ𝑧 𝐶 ∈ 𝐴
4		nfcsb1v 3876	. . . . 5 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
5	4	nfel1 2939	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴
6		csbeq1a 3866	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
7	6	eleq1d 2846	. . . 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 3434	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
10	9	imbi1i 351	. . . . 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 277	. . . 4 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝐶 ∈ 𝐴)))
13	12	ralbii2 3103	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
14	8, 13	bitr3i 279	. 2 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 ∈ 𝐴))
15		rabn0 4342	. 2 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
16		reusv2lem5 5358	. . 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 1933	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝐶
18	4	nfeq2 2940	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19	6	eqeq2d 2772	. . . . . 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 633	. . . . . . 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 277	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ (𝜑 ∧ 𝑥 = 𝐶)))
24	23	rexbii2 3104	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
25	20, 24	bitr3i 279	. . . 4 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝑥 = 𝐶))
26	25	reubii 3375	. . 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 351	. . . . . . 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 277	. . . . . 6 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
31	30	ralbii2 3103	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
32	27, 31	bitr3i 279	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
33	32	reubii 3375	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
34	16, 26, 33	3bitr3g 315	. 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 608	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  ∃!wreu 3364  {crab 3413  ⦋csb 3852  ∅c0 4285

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-nul 5255  ax-pow 5321

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  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-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-nul 4286

This theorem is referenced by:  cdleme25dN  40944