Theorem dffix2 34207

Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
dffix2	⊢ Fix 𝐴 = ran (𝐴 ∩ I )

Proof of Theorem dffix2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3436	. . . 4 ⊢ 𝑥 ∈ V
2	1	elfix 34205	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
3	1	elrn 5802	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4		brin 5126	. . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥))
5		ancom 461	. . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥))
6	1	ideq 5761	. . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
7	6	anbi1i 624	. . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
8	4, 5, 7	3bitri 297	. . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
9	8	exbii 1850	. . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
10		breq1 5077	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥))
11	10	equsexvw 2008	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
12	3, 9, 11	3bitri 297	. . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
13	2, 12	bitr4i 277	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I ))
14	13	eqriv 2735	1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ∩ cin 3886   class class class wbr 5074   I cid 5488  ran crn 5590   Fix cfix 34137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-fix 34161

This theorem is referenced by:  fixssrn  34209