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Mirrors > Home > ILE Home > Th. List > idref | GIF version |
Description: TODO: This is the same
as issref 4980 (which has a much longer proof).
Should we replace issref 4980 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 2164 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) | |
2 | 1 | fmpt 5629 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅) |
3 | vex 2724 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3, 3 | opex 4201 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V |
5 | 4, 1 | fnmpti 5310 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 |
6 | df-f 5186 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅)) | |
7 | 5, 6 | mpbiran 929 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
8 | 2, 7 | bitri 183 | . 2 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
9 | df-br 3977 | . . 3 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
10 | 9 | ralbii 2470 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅) |
11 | mptresid 4932 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | |
12 | 3 | fnasrn 5657 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
13 | 11, 12 | eqtr3i 2187 | . . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
14 | 13 | sseq1i 3163 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅) |
15 | 8, 10, 14 | 3bitr4ri 212 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2135 ∀wral 2442 ⊆ wss 3111 〈cop 3573 class class class wbr 3976 ↦ cmpt 4037 I cid 4260 ran crn 4599 ↾ cres 4600 Fn wfn 5177 ⟶wf 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 |
This theorem is referenced by: (None) |
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