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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 35815 | . 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) | |
2 | inss1 4252 | . . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴 | |
3 | dmss 5926 | . . 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 4037 | 1 ⊢ Fix 𝐴 ⊆ dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3969 ⊆ wss 3970 I cid 5596 dom cdm 5699 Fix cfix 35791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-dm 5709 df-fix 35815 |
This theorem is referenced by: (None) |
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