Theorem dffix2 35871

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 3492	. . . 4 ⊢ 𝑥 ∈ V
2	1	elfix 35869	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
3	1	elrn 5918	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4		brin 5218	. . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥))
5		ancom 460	. . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥))
6	1	ideq 5877	. . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
7	6	anbi1i 623	. . . . . 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 1846	. . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
10		breq1 5169	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥))
11	10	equsexvw 2004	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
12	3, 9, 11	3bitri 297	. . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
13	2, 12	bitr4i 278	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I ))
14	13	eqriv 2737	1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∩ cin 3975   class class class wbr 5166   I cid 5592  ran crn 5701   Fix cfix 35801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-fix 35825

This theorem is referenced by:  fixssrn  35873