Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > fixssdm | Structured version Visualization version GIF version |
Description: The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
fixssdm | ⊢ Fix 𝐴 ⊆ dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fix 33217 | . 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
2 | inss1 4202 | . . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴 | |
3 | dmss 5764 | . . 3 ⊢ ((𝐴 ∩ I ) ⊆ 𝐴 → dom (𝐴 ∩ I ) ⊆ dom 𝐴) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ dom (𝐴 ∩ I ) ⊆ dom 𝐴 |
5 | 1, 4 | eqsstri 3998 | 1 ⊢ Fix 𝐴 ⊆ dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3932 ⊆ wss 3933 I cid 5452 dom cdm 5548 Fix cfix 33193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-dm 5558 df-fix 33217 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |