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Mirrors > Home > MPE Home > Th. List > Mathboxes > fixssrn | Structured version Visualization version GIF version |
Description: The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
fixssrn | ⊢ Fix 𝐴 ⊆ ran 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffix2 35652 | . 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) | |
2 | inss1 4227 | . . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴 | |
3 | 2 | rnssi 5942 | . 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴 |
4 | 1, 3 | eqsstri 4011 | 1 ⊢ Fix 𝐴 ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3943 ⊆ wss 3944 I cid 5575 ran crn 5679 Fix cfix 35582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-fix 35606 |
This theorem is referenced by: (None) |
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