Theorem dffix2 33368

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 3499	. . . 4 ⊢ 𝑥 ∈ V
2	1	elfix 33366	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
3	1	elrn 5824	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4		brin 5120	. . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥))
5		ancom 463	. . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥))
6	1	ideq 5725	. . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥)
7	6	anbi1i 625	. . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
8	4, 5, 7	3bitri 299	. . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
9	8	exbii 1848	. . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥))
10		breq1 5071	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥))
11	10	equsexvw 2011	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
12	3, 9, 11	3bitri 299	. . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
13	2, 12	bitr4i 280	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I ))
14	13	eqriv 2820	1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ∩ cin 3937   class class class wbr 5068   I cid 5461  ran crn 5558   Fix cfix 33298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-fix 33322

This theorem is referenced by:  fixssrn  33370