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Mirrors > Home > ILE Home > Th. List > cnvi | GIF version |
Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvi | ⊢ ◡ I = I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 4691 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 1682 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
5 | 4 | opabbii 3995 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
6 | df-cnv 4547 | . 2 ⊢ ◡ I = {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} | |
7 | df-id 4215 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2170 | 1 ⊢ ◡ I = I |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 class class class wbr 3929 {copab 3988 I cid 4210 ◡ccnv 4538 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: coi2 5055 funi 5155 cnvresid 5197 fcoi1 5303 ssdomg 6672 |
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