Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elfix | Structured version Visualization version GIF version |
Description: Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
elfix.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elfix | ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fix 33322 | . . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I ) | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I )) |
3 | elfix.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | eldm 5771 | . . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥) |
5 | brin 5120 | . . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥)) | |
6 | ancom 463 | . . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥)) | |
7 | vex 3499 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | ideq 5725 | . . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥) |
9 | eqcom 2830 | . . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
10 | 8, 9 | bitri 277 | . . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴) |
11 | 10 | anbi1i 625 | . . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
12 | 5, 6, 11 | 3bitri 299 | . . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
13 | 12 | exbii 1848 | . . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
14 | 4, 13 | bitri 277 | . 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
15 | breq2 5072 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
16 | 3, 15 | ceqsexv 3543 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴) |
17 | 2, 14, 16 | 3bitri 299 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 class class class wbr 5068 I cid 5461 dom cdm 5557 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-dm 5567 df-fix 33322 |
This theorem is referenced by: elfix2 33367 dffix2 33368 fixcnv 33371 ellimits 33373 elfuns 33378 dfrecs2 33413 dfrdg4 33414 |
Copyright terms: Public domain | W3C validator |