Theorem reusv2lem3 5293

Description: Lemma for reusv2 5296. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2lem3	⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

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

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem3

Step	Hyp	Ref	Expression
1		simpr 488	. . . 4 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
2		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝐵 ∈ V
3		nfeu1 2587	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
4	2, 3	nfan 1907	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
5		euex 2576	. . . . . . . 8 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
6		rexn0 4422	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
7	6	exlimiv 1938	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
8		r19.2z 4406	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
9	8	ex 416	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
10	5, 7, 9	3syl 18	. . . . . . 7 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
11	10	adantl 485	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
12		nfra1 3140	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝐵 ∈ V
13		nfre1 3225	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵
14	13	nfeuw 2592	. . . . . . . 8 ⊢ Ⅎ𝑦∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
15	12, 14	nfan 1907	. . . . . . 7 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
16		rsp 3127	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (𝑦 ∈ 𝐴 → 𝐵 ∈ V))
17	16	impcom 411	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18		isset 3421	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
19	17, 18	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
20	19	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21		rspe 3223	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
22	21	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
23	22	ancrd 555	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)))
24	23	eximdv 1925	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)))
25	24	imp 410	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))
26	20, 25	syldan 594	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))
27		eupick 2634	. . . . . . . . . 10 ⊢ ((∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
28	1, 26, 27	syl2an2 686	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
29	28	ex 416	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵)))
30	29	com3l 89	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → (𝑦 ∈ 𝐴 → 𝑥 = 𝐵)))
31	15, 13, 30	ralrimd 3139	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
32	11, 31	impbid 215	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
33	4, 32	eubid 2586	. . . 4 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
34	1, 33	mpbird 260	. . 3 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
35	34	ex 416	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
36		reusv2lem2 5292	. 2 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
37	35, 36	impbid1 228	1 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  ∃!weu 2567   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  Vcvv 3408  ∅c0 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199  ax-pow 5258

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-v 3410  df-dif 3869  df-nul 4238

This theorem is referenced by:  reusv2lem4  5294  eusv4  5299