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Mirrors > Home > ILE Home > Th. List > idref | GIF version |
Description: TODO: This is the same
as issref 4916 (which has a much longer proof).
Should we replace issref 4916 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 2137 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) | |
2 | 1 | fmpt 5563 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅) |
3 | vex 2684 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 3, 3 | opex 4146 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V |
5 | 4, 1 | fnmpti 5246 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 |
6 | df-f 5122 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) ⊆ 𝑅)) | |
7 | 5, 6 | mpbiran 924 | . . 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 3925 | . . 3 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
10 | 9 | ralbii 2439 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 〈𝑥, 𝑥〉 ∈ 𝑅) |
11 | mptresid 4868 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) | |
12 | 3 | fnasrn 5591 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
13 | 11, 12 | eqtr3i 2160 | . . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝑥〉) |
14 | 13 | sseq1i 3118 | . 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 1480 ∀wral 2414 ⊆ wss 3066 〈cop 3525 class class class wbr 3924 ↦ cmpt 3984 I cid 4205 ran crn 4535 ↾ cres 4536 Fn wfn 5113 ⟶wf 5114 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: (None) |
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