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Mirrors > Home > MPE Home > Th. List > idref | Structured version Visualization version GIF version |
Description: TODO: This is the same
as issref 5428 (which has a much longer proof).
Should we replace issref 5428 with this one? - NM 9-May-2016.
Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
idref | ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) | |
2 | 1 | fmpt 6289 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅) |
3 | opex 4859 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
4 | 3, 1 | fnmpti 5935 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 |
5 | df-f 5808 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅)) | |
6 | 4, 5 | mpbiran 955 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
7 | 2, 6 | bitri 263 | . 2 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
8 | df-br 4584 | . . 3 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
9 | 8 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅) |
10 | mptresid 5375 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | |
11 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | fnasrn 6317 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
13 | 10, 12 | eqtr3i 2634 | . . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
14 | 13 | sseq1i 3592 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
15 | 7, 9, 14 | 3bitr4ri 292 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 ran crn 5039 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: retos 19783 filnetlem2 31544 |
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