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Mirrors > Home > MPE Home > Th. List > idref | Structured version Visualization version GIF version |
Description: Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) |
Ref | Expression |
---|---|
idref | ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) | |
2 | 1 | fmpt 7111 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅) |
3 | opex 5464 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
4 | 3, 1 | fnmpti 6693 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 |
5 | df-f 6547 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅)) | |
6 | 4, 5 | mpbiran 706 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
7 | 2, 6 | bitri 275 | . 2 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
8 | df-br 5149 | . . 3 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
9 | 8 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅) |
10 | mptresid 6050 | . . . 4 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥) | |
11 | vex 3477 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | fnasrn 7145 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
13 | 10, 12 | eqtri 2759 | . . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
14 | 13 | sseq1i 4010 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
15 | 7, 9, 14 | 3bitr4ri 304 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 〈cop 4634 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 ran crn 5677 ↾ cres 5678 Fn wfn 6538 ⟶wf 6539 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 |
This theorem is referenced by: retos 21481 filnetlem2 35730 |
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