Theorem reuhyp 1581

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 1580.

Hypotheses

Ref	Expression
reuhyp.1	⊢ (x ∈ C → B ∈ C)
reuhyp.2	⊢ ((x ∈ C ∧ y ∈ C) → (x = A ↔ y = B))

Assertion

Ref	Expression
reuhyp	⊢ (x ∈ C → ∃!y ∈ C x = A)

Distinct variable group(s):   x,A   y,B   y,C   x,y

Proof of Theorem reuhyp

Step	Hyp	Ref	Expression
1		reuhyp.1	. . . . 5 ⊢ (x ∈ C → B ∈ C)
2		elisset 1354	. . . . 5 ⊢ (B ∈ C → B ∈ V)
3	1, 2	syl 12	. . . 4 ⊢ (x ∈ C → B ∈ V)
4		eueq 1427	. . . 4 ⊢ (B ∈ V ↔ ∃!y y = B)
5	3, 4	sylib 173	. . 3 ⊢ (x ∈ C → ∃!y y = B)
6		eleq1 1149	. . . . . . . 8 ⊢ (y = B → (y ∈ C ↔ B ∈ C))
7	6, 1	syl5bir 184	. . . . . . 7 ⊢ (y = B → (x ∈ C → y ∈ C))
8	7	com12 13	. . . . . 6 ⊢ (x ∈ C → (y = B → y ∈ C))
9		pm4.71r 482	. . . . . 6 ⊢ ((y = B → y ∈ C) ↔ (y = B ↔ (y ∈ C ∧ y = B)))
10	8, 9	sylib 173	. . . . 5 ⊢ (x ∈ C → (y = B ↔ (y ∈ C ∧ y = B)))
11		reuhyp.2	. . . . . . 7 ⊢ ((x ∈ C ∧ y ∈ C) → (x = A ↔ y = B))
12	11	exp 291	. . . . . 6 ⊢ (x ∈ C → (y ∈ C → (x = A ↔ y = B)))
13	12	pm5.32d 491	. . . . 5 ⊢ (x ∈ C → ((y ∈ C ∧ x = A) ↔ (y ∈ C ∧ y = B)))
14	10, 13	bitr4d 409	. . . 4 ⊢ (x ∈ C → (y = B ↔ (y ∈ C ∧ x = A)))
15	14	bieudv 1013	. . 3 ⊢ (x ∈ C → (∃!y y = B ↔ ∃!y(y ∈ C ∧ x = A)))
16	5, 15	mpbid 170	. 2 ⊢ (x ∈ C → ∃!y(y ∈ C ∧ x = A))
17		df-reu 1207	. 2 ⊢ (∃!y ∈ C x = A ↔ ∃!y(y ∈ C ∧ x = A))
18	16, 17	sylibr 175	1 ⊢ (x ∈ C → ∃!y ∈ C x = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203  Vcvv 1348

This theorem is referenced by:  zmax 4618  rebtwnz 4620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  df-v 1349

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