Theorem reusv2 5304

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 3384	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝜑}
2		nfcv 2977	. . . 4 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝐶 ∈ 𝐴
4		nfcsb1v 3907	. . . . 5 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶
5	4	nfel1 2994	. . . 4 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴
6		csbeq1a 3897	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑦⦌𝐶)
7	6	eleq1d 2897	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ ⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴))
8	1, 2, 3, 5, 7	cbvralfw 3437	. . 3 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝐶 ∈ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}⦋𝑧 / 𝑦⦌𝐶 ∈ 𝐴)
9		rabid 3378	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ 𝜑))
10	9	imbi1i 352	. . . . 5 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝐶 ∈ 𝐴) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐶 ∈ 𝐴))
11		impexp 453	. . . . 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 3163	. . 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 4339	. 2 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
16		reusv2lem5 5303	. . 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 2995	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = ⦋𝑧 / 𝑦⦌𝐶
19	6	eqeq2d 2832	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = ⦋𝑧 / 𝑦⦌𝐶))
20	1, 2, 17, 18, 19	cbvrexfw 3438	. . . . 5 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
21	9	anbi1i 625	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 = 𝐶))
22		anass 471	. . . . . . 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 3245	. . . . 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 3391	. . 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 3437	. . . . 5 ⊢ (∀𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = 𝐶 ↔ ∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}𝑥 = ⦋𝑧 / 𝑦⦌𝐶)
28	9	imbi1i 352	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑥 = 𝐶) ↔ ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶))
29		impexp 453	. . . . . . 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 3163	. . . . 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 3391	. . 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 600	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 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142  ⦋csb 3883  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210  ax-pow 5266

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-nul 4292

This theorem is referenced by:  cdleme25dN  37507