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Mirrors > Home > ILE Home > Th. List > idref | GIF version |
Description: TODO: This is the same
as issref 5011 (which has a much longer proof).
Should we replace issref 5011 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 2177 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) | |
2 | 1 | fmpt 5666 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅) |
3 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3, 3 | opex 4229 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V |
5 | 4, 1 | fnmpti 5344 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 |
6 | df-f 5220 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅)) | |
7 | 5, 6 | mpbiran 940 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
8 | 2, 7 | bitri 184 | . 2 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
9 | df-br 4004 | . . 3 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
10 | 9 | ralbii 2483 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅) |
11 | mptresid 4961 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | |
12 | 3 | fnasrn 5694 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
13 | 11, 12 | eqtr3i 2200 | . . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
14 | 13 | sseq1i 3181 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
15 | 8, 10, 14 | 3bitr4ri 213 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 〈cop 3595 class class class wbr 4003 ↦ cmpt 4064 I cid 4288 ran crn 4627 ↾ cres 4628 Fn wfn 5211 ⟶wf 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 |
This theorem is referenced by: (None) |
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