Theorem dffix2 32601

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 3401	. . . 4 ⊢ 𝑥 ∈ V
2	1	elfix 32599	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
3	1	elrn 5612	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4		brin 4938	. . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥))
5		ancom 454	. . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥))
6	1	ideq 5520	. . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
7	6	anbi1i 617	. . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
8	4, 5, 7	3bitri 289	. . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
9	8	exbii 1892	. . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
10		breq1 4889	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥))
11	1, 10	ceqsexv 3444	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
12	3, 9, 11	3bitri 289	. . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
13	2, 12	bitr4i 270	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I ))
14	13	eqriv 2775	1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )

Syntax hints:   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107   ∩ cin 3791   class class class wbr 4886   I cid 5260  ran crn 5356   Fix cfix 32531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-fix 32555

This theorem is referenced by:  fixssrn  32603