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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 31135 | . 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
2 | inss1 3795 | . . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴 | |
3 | dmss 5245 | . . 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 3598 | 1 ⊢ Fix 𝐴 ⊆ dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3539 ⊆ wss 3540 I cid 4948 dom cdm 5038 Fix cfix 31111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 df-fix 31135 |
This theorem is referenced by: (None) |
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