Theorem reusv2lem5 5373

Description: Lemma for reusv2 5374. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2lem5	⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))

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

Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem reusv2lem5

Step	Hyp	Ref	Expression
1		tru 1574	. . . . . . . . 9 ⊢ ⊤
2		biimt 363	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
3	1, 2	mpan2 703	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝑥 = 𝐶 ↔ ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4		ibar 537	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝑥 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
5	3, 4	bitr3d 284	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
6		eleq1 2851	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
7	6	pm5.32ri 585	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ (𝐶 ∈ 𝐴 ∧ 𝑥 = 𝐶))
8	5, 7	bitr4di 292	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
9	8	ralimi 3102	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
10		ralbi 3120	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)) → (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
11	9, 10	syl 18	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
12	11	eubidv 2614	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 → (∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶)))
13		r19.28zv 4467	. . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
14	13	eubidv 2614	. . 3 ⊢ (𝐵 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
15	12, 14	sylan9bb 518	. 2 ⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶)))
16	1	biantrur 539	. . . . 5 ⊢ (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
17	16	rexbii 3112	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶))
18	17	reubii 3378	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶))
19		reusv2lem4 5372	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶))
20	18, 19	bitri 278	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥∀𝑦 ∈ 𝐵 ((𝐶 ∈ 𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21		df-reu 3370	. 2 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))
22	15, 20, 21	3bitr4g 317	1 ⊢ ((∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ ∅) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  ∃!weu 2596   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-nul 5269  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-nul 4287

This theorem is referenced by:  reusv2  5374