Theorem reuunixfr 2912

Description: Change the variable x in the expression for "the unique A such that φ" to another variable y contained in expression B. Use reuhyp 2911 to eliminate the last hypothesis.

Hypotheses

Ref	Expression
reuunixfr.1	⊢ (z ∈ C → ∀y z ∈ C)
reuunixfr.2	⊢ (y ∈ A → B ∈ A)
reuunixfr.3	⊢ (∪{y ∈ A∣ψ} ∈ A → C ∈ A)
reuunixfr.4	⊢ (x = B → (φ ↔ ψ))
reuunixfr.5	⊢ (y = ∪{y ∈ A∣ψ} → B = C)
reuunixfr.6	⊢ (x ∈ A → ∃!y ∈ A x = B)

Assertion

Ref	Expression
reuunixfr	⊢ (∃!x ∈ A φ → ∪{x ∈ A∣φ} = C)

Distinct variable groups:   x,B   x,z,C   x,y,A,z   φ,y,z   ψ,x,z

Proof of Theorem reuunixfr

Step	Hyp	Ref	Expression
1		reuunixfr.2	. . . . 5 ⊢ (y ∈ A → B ∈ A)
2		reuunixfr.6	. . . . 5 ⊢ (x ∈ A → ∃!y ∈ A x = B)
3		reuunixfr.4	. . . . 5 ⊢ (x = B → (φ ↔ ψ))
4	1, 2, 3	reuxfr 2910	. . . 4 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)
5		reucl2 2894	. . . . 5 ⊢ (∃!y ∈ A ψ → ∪{y ∈ A∣ψ} ∈ {y ∈ A∣ψ})
6		reucl 2891	. . . . . 6 ⊢ (∃!y ∈ A ψ → ∪{y ∈ A∣ψ} ∈ A)
7		hbrab1 1775	. . . . . . . 8 ⊢ (z ∈ {y ∈ A∣ψ} → ∀y z ∈ {y ∈ A∣ψ})
8	7	hbuni 2513	. . . . . . 7 ⊢ (z ∈ ∪{y ∈ A∣ψ} → ∀y z ∈ ∪{y ∈ A∣ψ})
9		reuunixfr.1	. . . . . . 7 ⊢ (z ∈ C → ∀y z ∈ C)
10		reuunixfr.5	. . . . . . 7 ⊢ (y = ∪{y ∈ A∣ψ} → B = C)
11	8, 9, 1, 3, 10	rabxfr 2908	. . . . . 6 ⊢ (∪{y ∈ A∣ψ} ∈ A → (C ∈ {x ∈ A∣φ} ↔ ∪{y ∈ A∣ψ} ∈ {y ∈ A∣ψ}))
12	6, 11	syl 10	. . . . 5 ⊢ (∃!y ∈ A ψ → (C ∈ {x ∈ A∣φ} ↔ ∪{y ∈ A∣ψ} ∈ {y ∈ A∣ψ}))
13	5, 12	mpbird 196	. . . 4 ⊢ (∃!y ∈ A ψ → C ∈ {x ∈ A∣φ})
14	4, 13	sylbi 199	. . 3 ⊢ (∃!x ∈ A φ → C ∈ {x ∈ A∣φ})
15		ax-17 973	. . . . 5 ⊢ (z ∈ C → ∀x z ∈ C)
16		hbrab1 1775	. . . . . . 7 ⊢ (z ∈ {x ∈ A∣φ} → ∀x z ∈ {x ∈ A∣φ})
17	15, 16	hbel 1569	. . . . . 6 ⊢ (C ∈ {x ∈ A∣φ} → ∀x C ∈ {x ∈ A∣φ})
18	17	a1i 8	. . . . 5 ⊢ (C ∈ A → (C ∈ {x ∈ A∣φ} → ∀x C ∈ {x ∈ A∣φ}))
19		eleq1 1537	. . . . 5 ⊢ (x = C → (x ∈ {x ∈ A∣φ} ↔ C ∈ {x ∈ A∣φ}))
20	15, 18, 19	reuuni2f 2889	. . . 4 ⊢ ((C ∈ A ⋀ ∃!x ∈ A x ∈ {x ∈ A∣φ}) → (C ∈ {x ∈ A∣φ} ↔ ∪{x ∈ A∣x ∈ {x ∈ A∣φ}} = C))
21		reuunixfr.3	. . . . . 6 ⊢ (∪{y ∈ A∣ψ} ∈ A → C ∈ A)
22	6, 21	syl 10	. . . . 5 ⊢ (∃!y ∈ A ψ → C ∈ A)
23	4, 22	sylbi 199	. . . 4 ⊢ (∃!x ∈ A φ → C ∈ A)
24		rabid 1772	. . . . . . 7 ⊢ (x ∈ {x ∈ A∣φ} ↔ (x ∈ A ⋀ φ))
25	24	baibr 688	. . . . . 6 ⊢ (x ∈ A → (φ ↔ x ∈ {x ∈ A∣φ}))
26	25	reubiia 1784	. . . . 5 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A x ∈ {x ∈ A∣φ})
27	26	biimp 151	. . . 4 ⊢ (∃!x ∈ A φ → ∃!x ∈ A x ∈ {x ∈ A∣φ})
28	20, 23, 27	sylanc 473	. . 3 ⊢ (∃!x ∈ A φ → (C ∈ {x ∈ A∣φ} ↔ ∪{x ∈ A∣x ∈ {x ∈ A∣φ}} = C))
29	14, 28	mpbid 195	. 2 ⊢ (∃!x ∈ A φ → ∪{x ∈ A∣x ∈ {x ∈ A∣φ}} = C)
30	24	baib 687	. . . 4 ⊢ (x ∈ A → (x ∈ {x ∈ A∣φ} ↔ φ))
31	30	rabbii 1808	. . 3 ⊢ {x ∈ A∣x ∈ {x ∈ A∣φ}} = {x ∈ A∣φ}
32	31	unieqi 2515	. 2 ⊢ ∪{x ∈ A∣x ∈ {x ∈ A∣φ}} = ∪{x ∈ A∣φ}
33	29, 32	syl5eqr 1524	1 ⊢ (∃!x ∈ A φ → ∪{x ∈ A∣φ} = C)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956   = wceq 958   ∈ wcel 960  ∃!wreu 1650  {crab 1651  ∪cuni 2507

This theorem is referenced by:  reuunineg 6068

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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