Theorem reusv2lem3 5192

Description: Lemma for reusv2 5195. (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 485	. . . 4 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
2		nfv 1892	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝐵 ∈ V
3		nfeu1 2635	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
4	2, 3	nfan 1881	. . . . 5 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
5		euex 2622	. . . . . . . 8 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
6		rexn0 4368	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
7	6	exlimiv 1908	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝐴 ≠ ∅)
8		r19.2z 4354	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
9	8	ex 413	. . . . . . . 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 482	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
12		nfra1 3186	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝐵 ∈ V
13		nfre1 3269	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑥 = 𝐵
14	13	nfeu 2639	. . . . . . . 8 ⊢ Ⅎ𝑦∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵
15	12, 14	nfan 1881	. . . . . . 7 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
16		rsp 3172	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (𝑦 ∈ 𝐴 → 𝐵 ∈ V))
17	16	impcom 408	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18		isset 3449	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
19	17, 18	sylib 219	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
20	19	adantrr 713	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21		rspe 3267	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
22	21	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
23	22	ancrd 552	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)))
24	23	eximdv 1895	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)))
25	24	imp 407	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))
26	20, 25	syldan 591	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵))
27		eupick 2688	. . . . . . . . . 10 ⊢ ((∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ∧ 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
28	1, 26, 27	syl2an2 682	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → 𝑥 = 𝐵))
29	28	ex 413	. . . . . . . 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 3185	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
32	11, 31	impbid 213	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
33	4, 32	eubid 2633	. . . 4 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
34	1, 33	mpbird 258	. . 3 ⊢ ((∀𝑦 ∈ 𝐴 𝐵 ∈ V ∧ ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
35	34	ex 413	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
36		reusv2lem2 5191	. 2 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
37	35, 36	impbid1 226	1 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  ∃!weu 2611   ≠ wne 2984  ∀wral 3105  ∃wrex 3106  Vcvv 3437  ∅c0 4211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-nul 5101  ax-pow 5157

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3862  df-nul 4212

This theorem is referenced by:  reusv2lem4  5193  eusv4  5198